iMTDODUCTION TO 
ACTUARIAL SCIENCE 




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INTRODUCTION TO 
ACTUARIAL SCIENCE 



By 
HARRY ANSON FINNEY 



AMERICAN INSTITUTE OF ACCOUNTANTS 
NEW YORK 



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Copyright, 1920, 
By Harry Anson Finney 



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INTRODUCTORY NOTE 

Part of the text of Introduction to Actuarial 
Science appeared originally in The Journal of 
Accountancy for November and December, 191 9. 

The copyright was donated by the author to 
the endowment fund of the American Institute 
of Accountants, and, in response to suggestions, 
the author added solutions of actuarial problems 
which had appeared in the examinations of the 
Institute. 

The text is published in book form under the 
endowment fund of the Institute. 

There has long been a demand for a clear, 
elementary text on actuarial science, and it is 
hoped that the demand will be fully met by this 
volume. 

The American Institute of Accountants is 
under a debt of gratitude to Mr. Finney for 
preparing the matter contained in this book 
and for his generous donation of the copyright 
to the endowment fund. 

A. P. RICHARDSON, 

Secretary. 
New York, June, 1920. ! 



TABLE OF CONTENTS 

Page 
General Definitions: 

Interest 2 

Rate 2 

Frequency 3 

Time 4 

Single Investments: 

Amount of i 5 

Compound Interest 8 

Computing Interest 9 

Computing Unknown Rate 9 

Computing Unknown Time. 9 

Present Value of i 10 

Compound Discount 12 

Annuities: 

Definition 13 

Amount of Ordinary Annuity 14 

Amount of Annuity Due 18 

Sinking Fund Contributions 21 

Amount of Annuity; Interest at Interim Dates 24 

Computing Unknown Rate 27 

Computing Unknown Time 29 

Present Value of Ordinary Annuity 30 

Present Value of Annuity Due 34 

Present Value of Annuity with Interim Interest 

Dates 36 

Present Value of Deferred Annuity 37 

Rents Produced by Known Present Value 39 

Equal Instalments in Payment of Debt and Interest 40 

V 



VI TABLE OF CONTENTS 

Page 
Leasehold Premiums: 

Premium Computed 42 

Premium Amortized 43 

Bonds: 

Prices 43 

Table 44 

Premium Computed 45 

Premium Amortized 47 

Discount Computed 49 

Discount Amortized 50 

Purchases at Intermediate Dates 52 

Optional Redemption 54 

Computing Effective Rate 58 

Depreciation: 

Annuity Method 61 

Sinking Fund Method 63 

Problems and Solutions: 

Depreciation 65 

Sinking Fund and Serial Redemption Contrasted . . 68 

Sinking Fund : 73 

Present Value and Amount of Annuity 74 

Sinking Fund 77 

Equal Instalments in Payment of Debt 81 

Bond Price 83 

Bond Price, Redeemable at Premium 84 

Leasehold Premium 85 

Present Value of Annuity 89 

Effective Rate on Bond 90 

Sinking Fund 93 



INTRODUCTION TO ACTUARIAL 
SCIENCE 



Introduction to Actuarial Science 

In the more comprehensive meaning of the 
term, actuarial science includes an expert knowl- 
edge of the principles of compound interest as 
well as the laws of insurance probabilities. Pub- 
lic accountants, however, are usually interested 
only in the interest phases of actuarial science, 
leaving the application of the laws of insurance 
probabilities to the actuary, who ascertains the 
measurement of risks and establishes tables of 
rates. This discussion of actuarial science will, 
therefore, be restricted to the phases thereof 
which deal with compound interest. 

There seems to be a more or less prevalent 
belief that the exacting of compound interest is 
illegal, and that this illegality makes the mathe- 
matics of compound interest an impractical 
matter of purely academic importance. While it 
may be illegal in some jurisdictions for a creditor 
to charge the debtor interest on unpaid interest, 
there can be no legal restriction against the col- 
lection and reinvestment of the interest. In fact, 
the mathematical theory of investment is based 
on the assumption that all interest accretions 

1 



2 INTRODUCTION TO ACTUARIAL SCIENCE 

become themselves a part of the investment, 
being converted at periodical intervals into inter- 
est-earning principal. The compound interest 
basis is the only scientific one where the accumu- 
lation or reduction of an investment extends over 
a series of periods. It is with compound interest, 
therefore, and not with simple interest, that 
actuarial science deals. 

Interest is the increase in investment or in- 
debtedness caused by the use of money or credit. 
The rapidity and extent of the increase depend 
on the factors of interest, which are rate, fre- 
quency and time. 

The rate is usually expressed in terms of per- 
centage and measures the fraction by which the 
investment is increased at each date of conversion 
of interest. Thus, a rate of 5% per period indi- 
cates that the interest each period will be .05 of 
the investment at the beginning of the period; 
or, stated in another way, the investment at the 
end of the period will be 1.05 times the invest- 
ment at the beginning of the period. Expressing 
the decimal interest rate by the symbol i (for 
instance, .05 = z), the interest earned during any 
period may be computed by multiplying the 
investment at the beginning of the period by i; 
and the increased investment at the end of each 
period may be computed by multiplying the 
investment at the beginning of the period by 



INTRODUCTION TO ACTUARIAL SCIENCE 3 

i+{. The symbol i+z is called the ratio of 
increase, because it measures the ratio existing 
between the investment at the beginning and 
the investment at the end of each period. 

The frequency is the length of the period in 
years, months or days between the dates of 
interest conversions. It is evident that the 
frequency of compounding will materially affect 
the rapidity with which an investment increases. 
For instance, an investment of ^i.oo for one year 
will amount to more if the loan is at iH% per 
period of 3 months than if at 6% per period of 
12 months. 

Increase during one year in investment of 
^i.cx) at 1^/2% per period of 3 months: 

1. 00 original investment 
Multiply by 1.015 ratio of increase 

1. 015 investment end of 3 months 
« 1.015 

1.03022s " " "6 " 

•« " 1. 01 5 

^ 1.045678 " " "9 " 
•• " i.ois 

' 1.061364 " " " 12 " 

Increase during one year in investment of ^i.oo 
at 6% per period of 12 months: 

1. 00 original investment 
Multiply by 1.06 ratio of increase 

1.06 investment end of 12 months 



4 INTRODUCTION TO ACTUARIAL SCIENCE 

One frequently sees interest tables in which 
the interest rates are stated as a certain per cent, 
per annum. It is better to express the rate as a 
certain per cent, per period, because, when the 
compounding occurs more frequently than once 
a year, the effective rate earned during a year is 
really greater than the nominal rate. Thus, in 
the foregoing illustration, the loan is made at 
iH% P^^^ period of three months. The custom- 
ary statement, however, is that the rate is 6% 
a year compounded quarterly. The nominal 
rate is 6%, but since the investment increases 
during the year from i.oo to i. 061364, the effec- 
tive rate per annum is 6.1364%. 

The time is the total number of periods over 
which the investment extends. It is customary 
in commercial parlance to state the time as a 
certain number of years, but as a matter of prin- 
ciple the time is a certain number of periods which 
may be of any duration, and the rate should, 
therefore, be stated as the rate per period. For 
instance, if money is lent at 6% per annum for 
4^/2 years, compounded semi-annually, the time 
(represented by the symbol n) is 9 periods, and 
the rate (represented by i) is 3% per period. 

Each dollar of an investment increases in the 
same ratio as every other dollar; therefore, in 
compound interest computations, it is customary 
to compute the required value on the basis of a 



INTRODUCTION TO ACTUARIAL SCIENCE 5 

principal of $1.00, and to multiply by the number 
of dollars in the principal. 

The Amount of i. 

Since interest increases the investment, the 
fundamental problem in interest is the computa- 
tion of the amount to which an investment will 
increase in a given time. It has already been 
noted that the increase depends upon i (the rate 
per period) and n (the number of periods). 
During each period the investment increases in 
the ratio of i+z. At 6% per period, an invest- 
ment of I will amount, at the end of i period, to 
1.06, or i+v during the next period the invest- 
ment will increase to 1.06, the investment at the 
beginning of the period, multiplied by 1.06, the 
ratio of increase, or to 1.1236, which is (i+z)^; 
at the end of the third period the investment will 
amount to 1.1236, the investment at the begin- 
ning of the period, multiplied by 1.06, or 1.191016, 
which is (i -\-iy. Or, stated generally, the invest- 
ment will amount at the end of n periods to 

This means that the ratio of increase is raised 
to the nth power, or that the amount is a product 
obtained by using the ratio of increase as a factor 
as many times as there are periods. Represent- 
ing the amount of i by S, the formula is 

S = (i+zT 



INTRODUCTION TO ACTUARIAL SCIENCE 



When compound interest tables are available, 
the amount can be determined by reference to 
them. A table of amounts appears as follows: 





Amount 


OF I AT Compound Interest 




Periods 


3% 


3^% 


4% 


4>^% 


5% 


I 


1.030000 


1.035000 


1.040000 


1.045000 


1.050000 


2 


1.060900 


1.071225 


1.08 1600 


1.092025 


1. 102500 


3 


1.092727 


1.108718 


1. 1 24864 


1.141166 


1. 157625 


4 


1.125509 


1. 147523 


1. 169859 


1.192519 


1. 215506 


5 


I.I59274 


1. 1 87686 


1. 216653 


1. 246182 


1.276282 


6 


I. 194052 


1.229255 


1.2653 19 


1.302260 


1.340096 


7 


1.229874 


1.272279 


1.315932 


1.360862 


1. 407100 


8 


1.266770 


1.3 16809 


1.368569 


1.422101 


1.47745s 



.012837 

8 

.102696, which is 



When interest tables are not available, the 
amount is easily computed by logarithms. To 
illustrate: what is the amount of i at 6% com- 
pound interest for 4 years, compounded semi- 
annually? 

S = i.o3^ 
The logarithm of 1.03 is 
To raise to the 8th power multiply by 
The product is 
the logarithm of 1.266764, or 1.03^. 
An interest table states this amount as 1.266770 

When neither an interest. table nor a table of 
logarithms is available, the amount may be com- 
puted by repeated multipHcations. The required 
amount is the product obtained by using the 
ratio of increase as a factor as many times as 
there are periods. Thus, 1.03^ may be computed 
as follows: 



INTRODUCTION TO ACTUARIAL SCIENCE 

1.03 

1.03 



1.0609 amount at end of 2 periods 

1.03 



1.092727 

1.03 

1. 125509 
1.03 

1.159274 
1.03 

I.194052 
1.03 

1.229874 
1.03 

1.266770 



This work can be materially reduced by recog- 
nizing the principle that the multiplication of 
any two powers of a number results in a power 
represented by an exponent equal to the sum of 
the exponents of the powers multipHed. For 
instance, i .0609 = i .03 2. 

Now since 1.0609 contains 1.03 twice as a 
factor, the product of 1.0609 multiplied by 1.0609 
will contain 1.03 four times as a factor. Thus 

1.0609 X 1.0609 = 1.125509 = 1.03* 
The eighth power can be obtained thus: 
1 . 125509 the 4th power 
multiplied by 1. 125509 "4 " 

1.266770 "8 



8 INTRODUCTION TO ACTUARIAL SCIENCE 

The seventh power can be obtained by 
multiplying any powers the sum of whose 
exponents is 7. 

Thus: Or: 

1.092727 the 3rd power 1.0609 the 2nd power 

1.125509 " 4th " 1. 159274 " 5th " 

1.229874 " 7th " 1.229874 " 7th " 

This principle may be appHed when it is 
desired to determine an amount beyond the scope 
of an interest table. Thus, if an interest table 
extends only to 20 periods, the amount for, say, 
75 periods, can be computed by multiplication of 
the amounts for periods within the scope of the 
table, as follows: 

20th X 20th X 20th X 1 5th = 75th 

Or the principle may be applied when an 
amount must be computed with no table avail- 
able. For instance, {i + iy^ may be computed 
thus: 

(l+z) x(i+0 =(i+iV 

(i+iVx(i+tV=(i+0* 

(l+l)8x(l + t)8=(l + ty« 

The Compound Interest 

Since the interest increases the investment, 
the difference between i and the amount of i 



INTRODUCTION TO ACTUARIAL SCIENCE 9 

is the compound interest. Representing the 
compound interest by the symbol /, 

/ = S-i 

Illustration: required the compound interest 
on I at 6% per annum for 4 years, compounded 
semi-annually? 

The amount of 1.03^ has already been com- 
puted as 1.266770. 

Then 1.266770—1 = .266770 the compound 
interest. 

The Rate 

When the investment, the amount and the 
time are known, the rate can be computed by 
logarithms. 

Illustration: if $80.00 invested at an unknown 
rate, compounded annually will amount to 
$107.20 in six years, what is the rate-f* 

80 X (i+z)^= 107.20 

Hence (1+0^=107.20 -5-80=1.34 

Since 1.34 is the 6th power of i-\-iy it is necessary to extract 
the 6th root, which is accompUshed as follows: 

log. 1.34=^127105 

log. ^Vi-34=-i27i05-T-6 = .02ii84 
.021184 is almost the exact logarithm of 1.05, or l+i 
Hence i = .os the rate. 

The Time 

When the investment, amount and rate are 
known, the time can be computed by logarithms. 



lO INTRODUCTION TO ACTUARIAL SCIENCE 

Illustration: for how many years must ^2,- 
ooo.oo remain at 5% interest compounded an- 
nually to produce ^5,054.00? 

Since 5054=2000x1.05^ 

5054-7-2000 =1.05'' 
Or 2.527=1.05^ 
log. 2. 5 27 = .402605 
log. 1.05 = .021 189 
.402605 -I- .02 1 1 89 = 19, the number of years. 

Present Value of i 

The present value of a sum due at a fixed 
future date is a smaller sum which, with interest, 
will amount to the future sum. When the time 
is more than one period, the smaller sum will 
accumulate at compound interest, increasing each 
period in the ratio of increase of i+z. 

Representing the present value by the symbol P 
the accumulated sum at the end of i period will be Px (i+O 
" " " " "2 periods " " Px(i+02 
" " " " "n " " "Px(i+z> 

Where the future sum is i 

Px(i+z> = i 
and since (i+i)''=S 

PxS = i 
or P = I-^S 

That is, the present value of i due n periods 
hence at a given rate may be computed by divid- 
ing I by the amount of i for n periods at the same 
rate. 



INTRODUCTION TO ACTUARIAL SCIENCE II 

Illustration: what is the present value of i 
due four periods hence at 3%? 

The table of amounts of i at 3% shows that 1.03^=1.125509. 
Then I ^1.125509 = . 888487, which is the present value of i 
at 3% for 4 pt riods as shown in the following illustrative table: 





Present Value of i 






Periods 


3% 


3^% 


4% 


4>^% 


5% 


I 


.970874 


.966184 


.961538 


.956938 


.952381 


2 


.942596 


.933511 


.924556 


.915730 


.907029 


3 


.915142 


.901943 


.888996 


.876297 


.863838 


4 


.888487 


.871442 


.854804 


.838561 


.822702 


5 


.862609 


.841973 


.821927 


.802451 


.783526 


6 


.837484 


.813501 


.790315 


.767896 


.746215 


7 


.813092 


.785991 


.759918 


.734828 


.710681 


8 


.789409 


.759412 


.730690 


.703185 


.676839 



When a table of present values is not available 
the value of P can be computed by dividing i by 
the value of S for the same time and rate, shown 
by a table of amounts of i. 

In the absence of any table, the value of P 
can be computed in one of the following ways: 

Required: the present value of i due 4 peri- 
ods hence at 3%. 

Compute the value of S, which in this case is 
1.03^ and divide i by the value of S. 

1.032=1.0609 

1.03^= 1.06092= 1. 125509 

1-7-1.125509=. 888487 

Or, use the ratio of increase as a divisor as many 
times as there are periods, using i as the first 



12 INTRODUCTION TO ACTUARIAL SCIENCE 

dividend, and each quotient as the dividend in 
the succeeding division; thus: 

I -M. 03 =.970874 present value of i due i period hence 
.970874-7- 1.03 = .942596 " " " I " 2 periods " 

.942596-M.03 =.915142 " " " I " 3 

.9151424-1.03 =.888487 " " " I " 4 

Or, divide i by the ratio of increase, and raise the 
quotient to the nth power; thus: 

I -M. 03 =.970874 
.970874 X .970874 = .942596 present value of i due 2 periods hence 
.942596 X. 942596 = .888487 " " "i "4 

Compound Discount 

The present value of i is the sum which will 
accumulate to i in a given time at a given rate. 
The difference between i and the present value 
of I is the compound discount, which will be 
represented by the symbol D. The compound 
discount can be computed by the formula: 

Illustration: required the compound discount 
on I due 4 periods hence at 3%. 

The value of P for 4 periods at 3% was computed above. It 
is .888487. 

Therefore i—. 888487 = . 111513, the compound discount. 

This is the formula to use when one has a 
table of present values. If one has only a table 
of amounts of i, it is better to use the formula: 



INTRODUCTION TO ACTUARIAL SCIENCE 1 3 

This formula requires explanation. The com- 
pound discount is really the compound interest 
earned on the present value. It is the interest 
which increases P to i. For instance, if one lends 
.888487 for 4 periods at 3%, the interest will be 
.III 51 3, increasing the investment to i. 

Now the interest earned is proportionate to 
the sum invested. If the investment is i, the 
compound interest will be /. If the investment 
is half of I, the compound interest will be Yi of ^• 
If the investment is .888487, the interest will be 
/x. 888487. That is, if the investment is P the 
earning of D will be /x P. But P is i -r-S. 

Hence the formula: 





Z)=/xP 


may be stated 


£>=/x(i^S) 


or 


D=/xi 


or 


Z)=/4-S 



To illustrate the application of this formula: 
required the compound discount on i for 4 
periods at 3%. Given: 1.03^=1.125509. 

Then .125509-M. 125509 = . 111513. 

Annuities 

A series of equal payments, running sup- 
posedly for more than a year and due at regular 
intervals, is an annuity. The intervals may be 
periods of any length, as a month, a quarter, a 



14 INTRODUCTION TO ACTUARIAL SCIENCE 

half year or a year. The periodical payments of 
an annuity are usually called rents. 

Since the rents may be invested when re- 
ceived, there is the problem of determining the 
amount to which the rents will accumulate. On 
the other hand it may be desired to compute the 
investment which, with interest accumulations, 
will permit the withdrawal of rents of stated 
amounts at stated intervals. The sum so invested 
is the present value of the annuity. These are 
the two fundamental annuity problems. The 
amount of an immediate annuity of i per period 
for n periods will be represented by the symbol 
Sn; the present value of an immediate annuity 
of I per period for n periods, by the sumbol a^. 

Amount of Annuity 

To illustrate the methods of computing the 
amount to which the rents of an annuity will 
accumulate, let it be assumed that a contract 
requires the following payments: 

Dec. 31, 1916 ^300.00 

" 31, 1917 300.00 

" 31, 1918 300.00 

" 3i> 1919 300.00 

Required the amount to which these payments 
will accumulate at December 31, 1919, if each 
rent is invested immediately at 3% per annum, 



INTRODUCTON TO ACTUARIAL SCIENCE 1 5 

Clearly the amount of the annuity will be 
the sum of the four ^300 rents plus the interest 
on each rent, as follows: 

Periods at Amount Amount 



Date 


Rent 


interest 


of I 


of 300 


ec. 31, 1916 


?3 00.00 


3 


1.092727 


327.8181 


" 3h 1917 


300.00 


2 


1.0609 


318.27 


" 3h 1918 


300.00 


I 


1.03 


309.00 


" 3i> 1919 


300.00 





1. 00 


300.00 




1200.00 


I255.088I 



Although the amount of an annuity may be 
computed by determining the amount of each 
rent and the sum of these amounts, it is unneces- 
sary to resort to this labor. 

The following short method may be used : 
To find the amount of an annuity of i for a 
given number of periods at a given rate, divide 
the compound interest on i for the number of 
periods at the given rate by the simple interest 
rate. Or, in symbols: 

Applied to the foregoing illustration: 

1.03^=1.125509 

Hence / = .125509 

And j-j = .•I25509-^ .03 =4.18363, the amount of an annu- 

ity of I 

4.18363 X 300=1255.089, the amount of an annu- 

ity of 30Q 



l6 INTRODUCTION TO ACTUARIAL SCIENCE 

The accumulation of this annuity may be 
shown thus: 

December 31, 191 6 Rent $300.00 

" 31, 1917 Interest at 3% on $300.00 9.00 

Rent 300.00 

Total $609.00 

" 31,1918 Interest at 3% on $609.00 18.27 

Rent 300.00 

Total $927.27 

" 31,1919 Interest at 3% on $927.27 27.82 

Rent 300.00 

Total $i>255.09 

To understand the formula j-, = /-M* it is 

»i ' 

necessary to consider the relation between com- 
pound interest and the amount of an annuity. 
To show this relation, we shall assume that the 
four $300.00 payments referred to in the preced- 
ing illustration represented the annual interest 
on $10,000.00 at 3%, the money being lent one 
year before the first interest payment was due. 
The four payments of $300.00 each, or $1,200.00 
altogether, constitute the simple interest at 3% 
on $10,000.00 for four years. By investing each 
$300.00 payment at 3%, the creditor makes his 
investment pay compound interest, because he 
earns interest on the interest. The sum of 
$1,255,089 is, therefore, the compound interest at 
3% for four periods on $10,000.00. It is also the 
amount of an annuity of four rents of $300.00 
e^ch^ accumulating at 3%, as shown by the two 



INTRODUCTION TO ACTUARIAL SCIENCE IJ 

foregoing solutions. Expressed formally as equa- 
tions, 

$1,255,089=/ on $10,000.00 (compound interest on $10,000.00). 

and also 

$1,255,089=/^ of $300.00 (amount of annuity of $300.) 
Dividing $1,255,089 by 10,000, 

$.1255089=/ on $1 
=j~j0f$.03 

Or, in other words, the compound interest on $1 
at 3% for four periods is equal to the amount of 
an annuity of ^.03 for four periods at 3%. Now, 
if .1255089 is the amount of an annuity of 3 
cents for four periods at 3%, .1255089-T-3 is the 
amount of an annuity of i cent for four periods 
^t 3%; and . 1 25 5089 -T- .03 is the amount of an 
annuity of $1 for four periods at 3%. Since 
.1255089 is also the compound interest on i for 
four periods at 3%, represented by /; and since 
.03 is the interest rate, represented by i. 



I-T-i^S: 



»l 



Interest books usually contain tables showing 
the amounts of annuities, in the following form: 

Amount of Annuity of i 



Periods 


3% 


3'A7o 


4% 


4K% 


5% 


I 


I.OOOOOO 


I.OOOOOO 


I.OOOOOO 


I.OOOOOO 


I.OOOOOO 


2 


2.030000 


2.035000 


2.040000 


2.045000 


2.050000 


3 


3.090900 


3.106225 


3.I2I600 


3.137025 


3.152500 


4 


4.183627 


4.214943 


4.246464 


4.278I9I 


4.3IOI25 


5 


5.309136 


5.362466 


5.416323 


5.470710 


5.525631 


6 


6.468410 


6.550152 


6.632975 


6.716892 


6.80I9I3 


7 


7.662462 


7.779408 


7.898294 


8.0I9I52 


8.142008 


8 


8.892336 


9.051687 


9.214226 


9.380014 


9.549109 



l8 iktroduction to actuarial science 

Amount of an Annuity Due 
The above formula for determining the 
amount of an annuity applies only to an ordinary 
or immediate annuity. It must be modified 
slightly to be applicable to an annuity due. 

An ordinary or immediate annuity is one 
whose rents are due at the ends of the periods. 
An annuity due is one whose rents are due at the 
beginnings of the periods. Since the last rent of 
an ordinary annuity is due at the end of the last 
period, no interest is earned after the payment 
of the last rent. But in the case of an annuity 
due, the last rent is paid in at the beginning of the 
last period, and hence all rents as well as the 
accumulated interest thereon bear interest for 
another period. 

The difference between these two classes of 
annuities may be made more apparent by the 
following table comparing an ordinary annuity, 
the rents being payable at the end of each period, 
and an annuity due, the rents being payable at 
the beginning of each period. 

Year Ordinary annuity Annuity due 

Rents due Rents due 

1916 January i 
December 31 

1917 January i 
December 31 

1918 January i 
December 31 

1919 January i 
December 31 



INTRODUCTION TO ACTUARIAL SCIENCE I9 

It is evident that if the rents and the interest 
rates of the two annuities are the same, the 
amount of the four rents of the ordinary annuity, 
at December 31, 1919, will be the same as the 
amount of the four rents of the annuity due at 
January i, 1919, because in each case there will 
be three interest accretions. But in the case of 
the annuity due, the entire amount accumulated 
at January i, 1919, will earn interest for a full 
period to December 31, 1919, the end of the last 
period. This will cause an increase computed by 
multiplying by the ratio of increase, i+z. Rep- 
resenting the amount of an annuity due of i per 
period for n periods by the symbol s-., the formula 
for the amount of an annuity due is: 

s^, = (/-^^)x(I+^) 

This means that the amount of an annuity due 
may be computed by determining the amount 
of an ordinary annuity of the same rents for the 
same number of periods and at the same rate and 
multiplying this amount by the ratio of increase. 
Illustration: if $300 is deposited on January 
I, 1916, 1917, 1918 and 1919, at 3%, what will 
be the amount of the annuity one year after 
making the last deposit .^ 

Si-, = (.i2SS09-^.03)Xi.03 

=4.18363X1.03 

= 4.309139, amount of annuity due of I 
4-309139X300= 1,292.74, amount of annuity due of 300. 



20 INTRODUCTION TO ACTUARIAL SCIENCE 

This method is a convenient one to employ 
when the amount of an ordinary annuity has 
been computed and it is desired to compute the 
amount of an annuity due of the same number of 
rents at the same rate per period. The amount 
of an annuity due for n periods may also be com- 
puted by determining the amount of an ordinary 
annuity of n+i periods and deducting one rent. 
Applying the method to the above illustration, 
the solution would be: 

First, if a table of annuity amounts is avail- 
able: 

The table shows the amount of an ordinary annuity of four 

rents of i at 3%, to be 5.309136. 
Then 5.309136—1=4.309136, the amount of an annuity 

due of four rents. 

Second, if a table of annuity amounts is not 
available: 

The amount of i at compound interest for five periods at 

3% may be computed, or ascertained from a table. It 

is 1. 1 59274. 
Then .159274-^.03=5.309133, the amount of an ordinary 

annuity of i for five periods. 
And 5.309133 — I =4.309133, the amount of an annuity due 

of I for four periods. 
And 4.309133X300=1,292.74, the amount of an annuity 

due of 300 for four periods. 

The accumulation of this annuity may be shown 
thus: 



INTRODUCTION TO ACTUARIAL SCIENCE 21 

January i, 1916 Rent 300.CXD 

January i, 191 7 Interest at 3% on 300.00 9.00 

Rent 300.00 

Total 609.00 

January i, 1918 Interest at 3% on 609.00 18.27 

Rent 300.00 

Total 927.27 

January i, 1919 Interest at 3% on 927.27 27.82 

Rent 300.00 

Total 1,255.09 

January i, 1920 Interest at 3% on 1,255.09 37.65 

Total 1,292.74 

Sinking Fund Contributions 
When the amount of each contribution to a 
sinking fund is computed on an actuarial basis, 
instead of on an arbitrary or a per-unit-of-output 
basis, the problem is to determine the periodical 
rents which will produce a required amount. The 
periodical contribution is computed by deter- 
mining the amount of an annuity of ^i for the 
given number of periods at the given rate. The 
amount of the required fund is then divided by 
the amount of an annuity of ^i. Since it is cus- 
tomary to deposit the periodical contributions 
at the ends of the periods, the divisor is usually 
the amount of an ordinary annuity. Represent- 
ing the total fund by the symbol S.F. and the 
periodical contributions by S.F.C.y 

Illustration: a company borrowed $25,000.00 
for four years, and provided for its repayment by 



22 INTRODUCTION TO ACTUARIAL SCIENCE 

establishing a sinking fund on a 3% basis, the 
contributions being made at the end of each of 
the four years. What was the amount of each 
contribution ? 

The amount of an ordinary annuity of i for 
four periods at 3% has already been computed at 
4.18363. 

Then S.F.C. = 25,ooo-r 4.18363 

= 5,975.67 

The following table shows the accumulation of 
the fund on a 3% basis: 

End of ist year: Sinking Fund Table 

Contribution S>975-67 

End of 2nd year: 

Interest at 3% on 5,975.67: 179-27 

Contribution 5>975-67 6,154.94 

Total 12,130.61 

End of 3rd year: 

Interest at 3%ton 12,130.61: 363.92 

Contribution 5,975.67 6,339.59 

Total 18,470.20 

End of 4th year: 

Interest at 3% on 18,470.20: 554-1 1 

Contribution 5>975-67 6,529.78 

Total 24,999.98 

Or the accumulation of the fund could be shown 

thus: Sinking Fund Table 

Year Interest Contributiofl Total 

1 5,975.67 5,975.67 

2 179.27 5,975.67 12,130.61 

3 363.92 5,975.67 18,470.20 

4 554.11 5,975.67 24,999.98 
Total 1,097.30 23,902.68 



INTRODUCTION TO ACTUARIAL SCIENCE 23 

If the contributions were made at the be- 
ginning of each period it would be necessary to 
divide the total fund by the amount of an annuity 
due. 

Illustration: referring to the preceding illus- 
tration, what would be the amount of each con- 
tribution if the company made the deposits at 
the beginning of each year? 

The amount of an annuity due of i for four periods at 3% has 
already been computed at 4.309139. 
Then 5./'.C. = 25,000-^-4.309139 
= 5,801.62 

The accumulation of the sinking fund may be 
tabulated as follows: 

Sinking Fund Table 
First Year: 

Beginning: Contribution 5,801.62 

End: Interest on 5,801.62 174.05 



Second Year: 




5>975-67 


Beginning: 


Contribution 


5,801.62 
11,777.29 


End: 


Interest on 11,777.29 


353-32 


Third Year 




12,130.61 


Beginning: 


Contribution 


5,801.62 
17,932.23 


End: 


Interest on 17,932.23 


537.97 


Fourth Year: 




18,470.20 


Beginning: 


Contribution 


5,801.62 


End: 


Interest on 24,271.82 


24,271.82 
728.16 



Total fund 24,999.98 



24 INTRODUCTION TO ACTUARIAL SCIENCE 

Or the accumulation may be shown as follows: 





Sinking Fund Table 




^ear 


Contribution 


Interest 


Total 


I 


5,801.62 


174.05 


5,975-67 


2 


5,801.62 


353.32 


12,130.61 


3 


5,801.62 


537-97 


8,470.20 


4 


5,801.62 


728.16 


24,999.98 



Total 23,206.48 i>793-50 

Interim Interest Dates 

Sometimes the interest is compounded more 
frequently than the rents are payable. For in- 
stance, the rents of the annuity may be due 
annually, while the interest may be compounded 
semi-annually or quarterly. Such a problem 
offers no difficulty if one remembers that the 
nominal and effective rates per annum differ 
under such conditions. 

Illustration: what is the amount of an 
annuity of four rents of ^100.00 each, payable 
at the expiration of a year, if the rents are in- 
vested at 6% per annum compounded semi- 
annually? 

Although the rate Js stated las 6% per annum, it is really 3% a 
half year. The ratio of increase each half year is, therefore, 1.03, 
and the ratio of increase each year is 1.032, or 1.0609. Hence the 
cflFective rate per annum is 6.09%. 



INTRODUCTION TO ACTUARIAL SCIENCE 2$ 

The amount of i for four years at 3% a half year, is 1.03', or 
1.266770; hence the compound interest is .266770. The amount 
of an annuity of i is computed as follows: 

Jjj = . 266770 -r .0609 
= 4.38046 

And the amount of an annuity of 100 is 438.05. 

The following table is set up by way of proof: 

Accumulation of Annuity 



100.00 



first Year: 




End: 


Contribution: 


Second Year: 




Middle: 


Interest on 100.00 


End: 


" " 103.00 




Contribution 




Total 


Third Year: 




Middle: 


Interest on 206.09 


End: 


" 212.27 




Contribution 




Total 


Fourth Year: 




Middle: 


Interest on 318.64 


End: 


" 328.20 




Contribution 




Total (as above) 



3.CX5 




3.09 


6.09 




100.00 




206.09 


6.18 




6.37 


12.SS 




100.00 




318.64 


9.56 




9.8s 


19.41 




100.00 



438.05 

If the rents are payable at the beginning of 
each year the amount is first computed for a 
similar annuity in which the rents are due at the 
end of each year, and this amount is multiplied 
by the ratio of increase. 



26 INTRODUCTION TO ACTUARIAL SCIENCE 



Illustration: Assuming that each of the 
^loo.oo rents in the preceding illustration was 
payable at the beginning of the year, what would 
be the amount of the annuity at the end of the 
fourth year? 

438.05 amount of annuity when rents are 
payable at end 
Multiply by 1.0609 

464.73 amount of annuity when rents are payable 
at beginning 

The following table is set up in proof: 

Accumulation of Annuity 



100.00 
3.00 
3.09 



First Year: 




Beginning: 


Contribution 


Middle: 


Interest on 100.00 


End: 


" 103.00 


Total 




Second Year: 




Beginning: 


Contribution 


Middle: 


Interest on 206.09 


End: 


" " 212.27 


Total 




Third Year: 




Beginning: 


Contribution 


Middle: 


Interest on 318.64 


End: 


" " 328.20 


Total 




Fourth Year: 




Beginning: 


Contribution 


Middle: 


Interest on 438.05 


End: 


" " 4SII9 


Total (as above) 



106.09 

100.00 

6.18 
6.37 

218.64 
100.00 

9.56 
9-85 

338-05 

100.00 

13.14 
13.54 

464.73 



INTRODUCTION TO ACTUARIAL SCIENCE 2/ 

The accumulation of interest at interim dates 
finds application in sinking funds which are 
accumulated by annual deposits, while the inter- 
est on the fund is compounded semi-annually. 
This is ordinarily the case with sinking funds, 
because the contributions are usually made 
annually, and the funds are invested in bonds 
bearing coupons payable semi-annually. To find 
the periodical contributions which will produce 
the required fund, compute the amount of an 
annuity of i by the process described in the pre- 
ceding paragraph and divide the required fund by 
this amount. 

Computing an Unknown Rate 

When the rents, the amount and the time of 
an annuity are known, it may be desired to 
compute the rate, as in the following illustra- 
tion: 

At what rate will an ordinary annuity of four 
rents of ^loo.oo amount to ^418.36? 

Problems of this nature are impossible of 
exact solution. Working with the formula for 
the amount of an annuity, 

n\ 

we know the value of j-,, which in this case is 
4.1836. The value of / is unknown, but substi- 
tuting for / its equivalent (i+z^ — I, in which 



28 INTRODUCTION TO ACTUARIAL SCIENCE 

the value of n is known to be 4, the knowns 
and unknowns can be stated as follows: 

4.1836= ^ Y 

This equation cannot be solved to obtain an 
exact value of i. Perhaps the best method of 
obtaining an approximation is by reference to a 
table showing amounts of annuities. A table 
shows the following amounts at various rates for 
four periods : 

Periods 2% 2y2% 3% 3K% 4% 

4 4.121608 4.152516 4.183627 4.214943 4.246464 

This table shows that the required rate is 3 %. But assume 
that the table does not show this rate, and that the amounts 
nearest to the 4.1836 stated in the problem are 

Rates Amounts 

at zyiJo 4.214943 

" 2K% 4.152516 

differences i % .062427 

Then a difference of 1% in the rate causes a difference of .062427 
in the amount. 

The amount at the unknown rate is 4.1836 

and the amount at the next lower known rate is 4. 15 25 16 
and the difference is .031084 

If an increase of .062427 in the amount is caused by an increase of 
1% in the rate, the increase of .031084 in the amount is caused by 
an increase in the rate of approximately 

.03 I 084 -T- .062427 = .498 

Hence the approximate rate is 2.5%+. 498%, or 2.998%, which is 
very close to the true rate of 3%. 



introduction to actuarial science 29 

Computing an Unknown Time 

When the rents, the amount and the rate of 
an annuity are known, the unknown number of 
periods can be computed by logarithms. 

Illustration: in how many periods will rents 
of $100.00 amount to $418.36 at 3%.?^ 

Dividing $418.36 by 100, the amount of an annuity of i is 
determined as 4.1836. 



Applying the formula j^, =/-^t, or 

i 



^,=(i±5!=l 



the knowns and unknowns are 



4.1836 



_ i.03° 



.03- 

Multiplying both sides by .03 4.1836 X.03 = 1.03°— i 
or .125508 = 1.03^—1 

adding i to both sides 1.125508 = 1.03° 

Then by logarithms: 

log. 1. 125508 =log. 103 Xn 

and |2i^_LHSS2?=„ 

log. 1.03 

log. 1. 125508 is .051346 
log. 1.03 is .012837 
.051346 -r- .012837 = 4, the number of periods. 

Since 1. 1 25508 is the amount of i for n periods 
at the known rate, and since 1.03 is the ratio of 
increase, i+i, the formula for computing the 
time required to produce a known amount of an 
annuity may be stated 

log. S 
^~log. (i+z) 



30 introduction to actuarial science 

Present Value of Annuity 

The present value of an annuity is the sum 
which must be put at interest to produce the 
desired rents. This invested sum is increased 
by the interest accretions and decreased by the 
payment of the rents, the balance remaining at 
the date when the last rent is due being exactly 
sufficient to provide for the last rent. 

Illustration: a contract requires the pay- 
ment of ^loo.oo on December 31, 1916, and each 
year thereafter until December 31, 1919. What 
sum invested at 3% per annum one year before 
the first ^100.00 payment is due will be sufficient 
to provide the four payments? 

Clearly the sum will be less than ^400.00 be- 
cause of the interest. The present value of each 
of the four rents may be computed, and the sum 
of these present values will be the present value 
of the annuity. Thus : 

Rent due Rent Periods present value Present value 







earns interest 


at 3% 


Dec. 31, 1916 


100.00 


I 


97.0874 


" 31, 1917 


100.00 


2 


94.2596 


" 31, 1918 


100.00 


3 


91.5142 


" 3i> 1919 


100.00 


4 


88.8487 



Present value of four rents 371-7099 

When the present value of an annuity is com- 
puted in this manner, the present value of the 



INTRODUCTION TO ACTUARIAL SCIENCE 3 1 

first rent may be computed by dividing the rent 
by the ratio of increase (i+i). This present 
value is divided by the ratio of increase to deter- 
mine the present value of the second rent; the 
quotient of each division serves as the dividend 
for the next succeeding division, thus: 

100.00 -M. 03 =97.0874 present value of rent due i period hence 
97.0874-5-1.03=94.2596 " " " " " 2 periods " 

94.2596-1.03=91.5142 " " " " " 3 " 

9i.5i42--i.03 = 88.8487 " " " " " 4 " 

371.7099 present value of the annuity 

This method is too laborious to be employed 
when there are many rents, and the labor can be 
avoided by applying the following short method : 

To compute the present value of an annuity 
of I for a given number of periods at a given rate, 
divide the compound discount on i for the same 
number of periods at the given rate by the inter- 
est rate. Representing the present value of an 
ordinary or immediate annuity of i per period 
for n periods by the symbol <^-., 

Applying this formula to the illustration, the first step is to 
determine the compound discount, D. A table of present values 
of 1 shows .888487 as the present value of i at 3% due four 
periods hence. 

Then i —.888487 = , 111513, the compound discount; and 
.111513-^.03 = 3.7171 the present value of an annuity of i; and 
3.7171X100 = 371.71 the present value of an annuity of 100. 



32 INTRODUCTION TO ACTUARIAL SCIENCE 

The reduction of this present value may be 
tabulated as follows: 

December 31, 1915 Present value 37171 

December 31, 1916 Interest at 3% on 371.71 11. 15 

Total 382.86 

Rent deducted 100.00 



Balance 282.86 

December 31, 1917 Interest at 3% on 282.86 8.49 



Total 291.35 

Rent deducted 100.00 



Balance I9I-35 

December 31, 1918 Interest at 3% on 191.35 5.74 

Total 197.09 

Rent deducted 100.00 



Balance 97*09 

December 31, 1919 Interest at 3% on 97.09 2.91 

Total 100.00 

Rent deducted 100.00 



Balance 0.00 

Or thus: 

Schedule of Reduction of Annuity 



Date 


Interest 


Rent 


Balance 




added 


deducted 




►ec. 31, 1915 






371.71 


" 31, 1916 


11.15 


100.00 


282.86 


" 31, 1917 


8.49 


100.00 


191.35 


" 31, 1918 


S.74 


100.00 


97.09 


" 3h 1919 


2.91 


100.00 


0.00 




28.29 


400.00 





INTRODUCTION TO ACTUARIAL SCIENCE 



33 



Interest books usually contain tables showing 
present values of annuities, in the following form: 





Present Value of Annuity 


OF I 




Periods 


3% 


3>^% 


4% 


^y^% 


5% 


I 


.970874 


.966184 


.961538 


.956938 


.952381 


2 


1.913470 


1.899694 


1.886095 


1.872668 


I.8594IO 


3 


2.82861 1 


2.801637 


2.775091 


2.748964 


2.723248 


4 


3.717098 


3.673079 


3.629895 


3.587526 


3.545951 


5 


4-579707 


4.515052 


4.451822 


4.389977 


4.329477 


6 


5417191 


5.328553 


5.242137 


5.157872 


5.075692 


7 


6.230283 


6. 1 14544 


6.002055 


5.892701 


5.786373 


8 


7.019692 


6.873956 


6.732745 


6.595886 


6.463213 



In the absence of such a table, the present 
V alue of an annuity can be computed by repeated 
divisions to determine the present value of each 
successive rent, as illustrated above, or by apply- 
ing the formula a-.=D-T-i, If a table of present 
values of i is available, the value of D can be 
obtained by subtraction, thus : 



i.oo 



.888487 present value of i at 3% for 4 periods. 



.11 15 13 compound discount at 3% for 4 periods. 

If one has only a table of amounts of i, the 
value of D can be computed by applying the 
formula D = I-^Sy thus: The table of amounts 
of I shows : 

1.03^=1.125509 

Then .125509-M. 125509 = . 111513 



34 introduction to actuarial science 

Present Value of Annuity Due 

If the rents are payable at the beginning of 
each period, no interest will be earned on the 
invested sum before withdrawing the first rent. 
Therefore it will be necessary to invest a larger 
sum than would be invested if the annuity were 
an ordinary one. If the present value of an 
ordinary annuity for the same number of periods 
is known, the present value of an annuity due 
may be computed by a method based on the 
following reasoning: 

The extra amount to be invested is equal to 
the interest earned on the present value of an 
ordinary annuity during the first period. There- 
fore, the present value of an annuity due can be 
computed by multiplying the present value of 
an ordinary annuity of the same number of 
rents at the same rate, by i+^ the ratio of 
increase. 

Illustration: what sum must be invested on 
January i, 1916, to permit the withdrawal of 
four rents of ^100.00 each on the first day of 
January, 1916, 1917, 1918 and 1919? Interest 
^t 3% a year. 

This is an annuity due of 4 rents at 3 %. The present value of 
an ordinary annuity of 4 rents of i at 3% was computed in the 
preceding illustration. It is 3.717099. Then 3.717099X1.03 = 
3.828612, the present value of an annuity due of i. And 3.828612 
X 100 = 382.86, the present value of an annuity due of 100. 



INTRODUCTION TO ACTUARIAL SCIENCE 35 

The reduction of this present value may be 
shown thus: 

January i, 1916 Present value $382.86 

Rent deducted 100.00 





Balance 


282.86 


January i, 1917 


Interest at 3% on $282.86 


8.49 




Total 


291.35 




Rent deducted 
Balance 


100.00 




191-35 


January i, 1918 


Interest at 3% on $191.35 


5.74 




Total 


197.09 




Rent deducted 
Balance 


100.00 




97.09 


January i, 1919 


Interest at 3% on $97.09 


2.91 




Total 


' 100.00 




Rent deducted 


100.00 



Balance 0.00 

The present value of the annuity due for four 
periods is the sum of the first rent plus the 
present value of the last three rents. Therefore, 
when a table of present values of ordinary annui- 
ties is available, take the present value of such 
an annuity for n — i periods and add one rent 
To illustrate: the annuity table shows 

2.82861 1 present value of ordinary annuity of 3 rents at 3% 
Add 1. 000000 



3.82861 1 present value of annuity due of 4 rents at 3% 



36 introduction to actuarial science 
Interim Interest Dates 

In cases where the time between the with- 
drawals of rents is divided into smaller interest 
periods, as in an annuity where the interest is 
compounded semi-annually and the rents are 
withdrawn , annually, consideration must be 
given to the difference between the nominal and 
the effective interest rates. 

Illustration: what sum invested at 6% a year, 
interest compounded semi-annually, will produce 
four rents of ^loo.oo, drawn annually, the first 
withdrawal being made one year after investing 
the present value of the annuity ? 

With eight interest periods and a rate of 3 % per period, the 
first step in the solution is to find the present value of i due 8 
periods hence at 3%. This is shown by a table of present values 
to be .789409. Then the compound discount is .210591. 
The effective rate per year is computed thus: 

1.03^ = 1.0609 ratio of increase per annum. 
And 6.09% is the effective rate per annum. 
Then .210591-^.0609 = 3.45798 present value of annuity of i. 
3.45798 X 100 = 345.80 present value of annuity of 100. 

The reduction of this annuity may be shown 
thus: 



345-80 
10.37 
10.69 
366.86 
100.00 
266.86 



First year: 




Beginning: 


Investment 


Middle: 


Interest at 3% on ^345.80 


End: 


" " " " 356.17 




Total 




Rent withdrawn 




Balance 



INTRODUCTION TO ACTUARIAL SCIENCE 37 



Second year: 
Middle: 
End: 


Interest at 3% on 
« «« (« <( 

Total 

Rent withdrawn 

Balance 

Interest at 3% on 

Total 
Rent withdrawn 

Balance 

Interest at 3% on 

(( « (( it 

Total 

Rent withdrawn 


266.86 
274.87 

183.12 
188.61 

94.27 
97.10 


8.01 

8.25 

283.12 
100.00 


Third year: 
Middle: 
End: 


183.12 

5.49 
5.66 




194.27 
100.00 


Fourth year: 
Middle: 
End: 


94.27 

2.83 
2.91 




100.01 
100.01 



Balance 



0.00 



Present Value of Deferred Annuity 

A deferred annuity is one which does not begin 
to run until after the expiration of a number of 
periods. For instance, an annuity of 5 rents 
deferred 10 periods is one which does not begin 
to run until after the expiration of 10 periods. If 
the periods are a year in length, ten years will 
elapse before the first period in which a payment 
is to be made, and as the rents are ordinarily due 
at the ends of periods, the first rent will not be 
due until the end of the eleventh year. 



38 INTRODUCTION TO ACTUARIAL SCIENCE 

To illustrate the method of computing the 
present value of such an annuity, assume that 
the 5 rents are $500 each, the annuity being de- 
ferred ten periods. The present value at 5% a 
year is desired. 

The rents are payable at the end of the nth, 12th, 13th, 14th 
and 15th years. If we were required to find the present value of the 
annuity at the beginning of the nth year, the solution would con- 
sist merely of computing the present value of an ordinary annuity 
of 5 rents, as follows: 

Present value of i at 5% in 5 periods =.783 5 26 
Compound discount .216474 

.216474 -ir .05 = 4.32948 present value of ordinary 
annuity of i. 

By investing 4.32948 at the beginning of the nth year it will be 
possible to draw out rents of i at each rent date. But since the 
annuity is deferred 10 periods the investment, made at the beginning 
of the first year, will earn interest for 10 periods before the beginning 
of the nth year. Then if 4.32948 invested at the beginning of the 
nth period will produce the annuity, the investment at the begin- 
ning of the first year need be only the present value of 4.32948. 

Present value of i at 5% in 10 periods = .613913 

Multiply by 4.32948 

Present value deferred annuity of i 2.65792 
Multiply by 500 

Present value of deferred annuity of 500 1328.96 

The method of procedure may be stated thus: 
to find the present value of an annuity of i for n 
periods deferred m periods, compute the value of 
an ordinary annuity of i for n periods and multi- 
ply by the present value of i due m periods hence. 
Proof of the accuracy of 1328.96: 



INTRODUCTION TO ACTUARIAL SCIENCE 39 

This investment earns compound interest for 
II years before the first rent is withdrawn: 

1.0511=1.710339 

Multiply by 1328.96 





2,272.97 






Deduct I St rent 


500.00 








1,772.97 


value end of nth year 


Add 5% interest 


88.65 
1,861.62 






Deduct 2nd rent 


500.00 


« 






1,361.62 


" "i2th " 


Add 5% interest 


68.08 








1,429.70 




Deduct 3rd rent 


500.00 


« 






929.70 


" "13th " 


Add 5% interest 


46.49 
976.19 






Deduct 4th rent 


500.00 


(( 






476.19 


" "14th " 


Add 5% interest 


23.81 








500.00 




Deduct 5th rent 


500.00 







« (( t( . 



15th " 

Rents Produced by Known Present Value 

An annuity problem of frequent appHcation 
is the determination of rents produced by a 
known present value. To illustrate: if ^5,000.00 



40 INTRODUCTION TO ACTUARIAL SCIENCE 

is invested at 5% a year, what rents can be with- 
drawn at the end of each of 10 years? 

If it were desired to withdraw rents of ^i, the present value of 
an annuity of 1 for 10 periods at 5% would be invested. This 
present value could be determined from an interest table, or 
computed. It is 7.721735. 

Then 5,000 -^ 7.72 173 5 = 647.52 annual rent produced by in- 
vestment of ^5,000. 

Hence to find the rents, divide the investment or 
known present value by the present value of an 
annuity of i for n periods at the rate i. 

Equal Instalments in Payment of a Debt 

The procedure explained above is used to 
compute the equal periodical amounts to be paid 
in settlement of the principal and interest of a 
debt. 

To illustrate: assume that A borrows ^5,000.00 
from B at 5% and agrees to pay principal and 
interest in ten equal annual instalments, the first 
instalment to be paid at the expiration of one 
year. These ten payments are an annuity, and 
if A invested the $5,000.00 at 5% it would, of 
course, exactly provide the annual amounts 
which he could draw out and pay to B. 

Since $5,000.00 is the present value of lo unknown frents 
at 5%, and since 7.721735 is the present value of lo rents of i 
at 5% 

5,000.00-^7.721735=647.52 the equal instalment. 



INTRODUCTION TO ACTUARIAL SCIENCE 4I 

The reduction of the debt may be shown as fol- 
lows: 

Annuity Repayment of Debt 







Payment 


Balance of 


ear 


Total 


Of interest 


On principal 


principal 
5000.00 


I 


647.52 


250.00 


397.52 


4602.48 


2 


647.52 


230.12 


417.40 


4185.08 


3 


647.52 


209.25 


438.27 


3746.81 


4 


647.52 


187.34 


460.18 


3286.63 


5 


647.52 


164.33 


483.19 


2803.44 


6 


647.52 


140.17 


507-35 


2296.09 


7 


647.52 


114.80 


532.72 


1763.37 


8 


647.52 


88.17 


559.35 


1204.02 


9 


647.52 


60.20 


587.32 


616.70 


10 


647.52 


30.83 


616.69 


.01 




6475.20 

t : : 3 


1475.21 


4999.99 





Leasehold Premiums 

When the rental value of real estate has in- 
creased, the holder of a lease may prefer to sublet 
rather than to continue to occupy the property. 
If so, he takes as his gain the difference between 
the rent he pays on the original lease and the 
rent he receives on the sub-lease. 

For instance, A owns property which he has 
leased to B for 20 years at ^3,000.00 a year, 
payable in advance. At the expiration of the 
seventh year C wishes to occupy the property 
and is willing to pay ^4,000.00 a year for it. 
Several different arrangements may be made. 



42 INTRODUCTION TO ACTUARIAL SCIENCE 

B may sub-lease to C, collecting ^4,000.00 a 
year, and paying ^3,000.00 to A. C may pay 
^3,000.00 to A and $1,000.00 to B each year. 
He may pay A $3,000.00 a year, and pay B 
the present value of the thirteen payments of 
$1,000.00 each. The present value of these 
$1,000.00 payments is the premium paid for 
the lease. 

Since the rent is payable in advance, the first 
rent is due. The premium to be paid is the pres- 
ent value of an annuity due of 13 rents. If B and 
C agree on 5% as the rate for discounting the 
annuity, the computation of the premium would 
Be as follows: 

.530321 = present value of i due 13 periods hence at 5% 
.469678-j- .05 =9.39356, present value or ordinary annuity of i 
9.39356X1000 = 9,393.56, present value of ordinary annuity of 

1000 
9>393-5^X 1.05 = 9,863.24, present value of annuity due of 1000. 

Or as follows: 

8.863252, present value of ordinary annuity of i at 5% for 12 

periods, as shown by annuity table. 
8.863252X1000 = 8,863.252 present value of ordinary annuity of 

1000 

8,863.25 + 1000 = 9,863.25 present value of annuity due of 1000 

for 13 periods. 

C would charge this 9,863.25 to leasehold 
premium, and would write it off as schedukd be- 
low: 





INTRODUCTION 


TO ACTUARIAL SCIENCE 43 




Reduction 


OF Leasehold Premium 




Year 


Debit 


Credit 


Credit 


Balance 




rent 


interest 


leasehold premium 


9,863.25 


I 


i,ooo.oo 




1,000.00 


8,863.25 


2 


i,ooo.oo 


443.16 


556.84 


8,306.41 


3 


i,ooo.oo 


415.32 


584.68 


7,721.73 


4 


i,ooo.oo 


386.09 


613.91 


7,107.82 


5 


i,ooo.oo 


355-39 


644.61 


6,463.21 


6 


1,000.00 


323.16 


676.84 


5,786.37 


7 


i,ooo.oo 


289.32 


710.68 


5,075-69 


8 


i,ooo.oo 


253.79 


746.21 


4,329.48 


9 


i,ooo.oo 


216.47 


783-53 


3,545-95 


10 


IjOOO.OO 


177.30 


822.70 


2,723.25 


II 


i,ooo.oo 


136.16 


863.84 


1,859.41 


12 


i,ooo.oo 


92.97 


907.03 


952.38 


13 


1,000.00 


47.62 


952.38 




Total 


13,000.00 


3,136.75 


9,863.25 





If the rents had been payable at the end of 
each year and the transfer of the lease had been 
made at the beginning of the year, the computa- 
tion of the premium would be a problem of the 
present value of an ordinary annuity instead of 
the present value of an annuity due. 

Bond Prices 

The price at which a bond will sell is affected 
by the nominal interest rate, the mortgaged 
security, the financial standing of the issuing 
company and the probability of being able to sell 
the bond if occasion requires. 

Bonds rarely sell on the market at par — usu- 
ally a premium is added or a discount deducted. 



44 INTRODUCTION TO ACTUARIAL SCIENCE 

Quotations are made ^'on a basis'^ or "at a price/' 
When bonds are sold at a price other than par the 
effective interest rate differs from the nominal or 
coupon rate. Thus if a bond is issued at a dis- 
count, the principal borrowed is really less than 
par. Moreover, the borrower pays not only the 
interest coupons but also the discount for the use 
of the borrowed money. Hence the effective rate 
on the loan is greater than the nominal rate. If 
the bond sells at a premium, the principal bor- 
rowed is more than par; and since the borrower 
does not have to pay back the premium at matur- 
ity, the premium is really a deduction from the 
interest. Hence the effective rate is less than the 
nominal rate. 

Quotations *'on a basis'' state the effective 
rate to be earned. The price, above or below 
par, may then be found in a bond table, or com- 
puted. Thus a 5 year 6% bond of ^loo.oo bought 
on a 5% basis would cost ^104.38 as shown in 
the following table — or a 5 year 5% bond of 
^100.00 bought on a 6% basis would cost ^95. 73. 

Illustration of a Bond Table 



Per cent. 








per 




5 years — Interest payable semi-annually 


annum 


3% 


^y2% 


4% 4>^% 5% 6% 7% 


4.7s 


92.29 


94.49 


96.70 98.90 loi.io 105.51 109.91 


4.80 


92.08 


94.28 


96.48 98.68 100.88 105.28 109.58 


4.87s 


91.77 


93.96 


96.16 98.35 100.55 104.94 109.33 


4.90 


91.66 


93.86 


96.05 98.25 100.44 104-83 109.21 



INTRODUCTION TO ACTUARIAL SCIENCE 45 

Illustration of a Bond Table — Com. 
Per cent. 5 years — Interest payable semi-annually 



per 
















annum 


3% 


3>^% 


4% 


4>^% 


5% 


6% 


7% 


5. 


91.25 


93.44 


95.62 


97.81 


100.00 


104.38 


108.75 


S.io 


90.83 


93.02 


95.20 


97.38 


99.56 


103.93 


108.29 


5.125 


90.73 


92.91 


95.09 


97.27 


99.45 


103.82 


108.18 


5.20 


90.42 


92.60 


94.78 


96.95 


99.13 


103.48 


107.84 


S.25 


90.22 


92.39 


94.57 


96.74 


98.91 


103.26 


107.61 


5.30 


90.01 


92.18 


94.35 


96.53 


98.70 


103.04 


107.38 


5.37s 


89.71 


91.87 


94.04 


96.21 


98.37 


102.71 


107.04 


5.40 


89.61 


91.77 


93.94 


96.10 


98.27 


102.60 


106.93 


5-50 


89.20 


91.36 


93.52 


95.68 


97.84 


102.16 


106.48 


5.625 


88.70 


90.85 


93.00 


95.16 


97.31 


101.61 


105.92 


5.75 


88.20 


90.34 


92.49 


94.63 


96.78 


101.07 


105.37 


5-875 


87.70 


89.84 


91.98 


94.12 


96.26 


100.53 


104.81 


6. 


87.20 


89.34 


91.47 


93.60 


95.73 


100.00 


104.27 



Computing the Premium 

When the effective rate is less than the nomi- 
nal rate, the premium may be computed by a 
method based on the following reasoning: 

Assume that the par of the bond is ^1,000.00, 
the time 5 years, the nominal rate 6% a year 
payable semi-annually, and that the bond is to 
be purchased on a 5% basis. The conditions are 
such that the purchaser is satisfied with 5% a 
year; therefore, if the bond bore coupons of 2^^% 
or ^25.00 payable each six months the bond would 
presumably sell at par. In other words, the 
purchaser would pay par, $1,000.00, for the right 
to receive par at maturity and interest of $25.00 
semi-annually. But since the coupons are $30.00 



46 INTRODUCTION TO ACTUARIAL SCIENCE 

each, he must also pay for the right to receive 
this extra ^5.00 each six months. This semi- 
annual payment of ^5.00 is an annuity, and the 
purchaser will pay its present value discounted 
at the effective interest rate of 2^^% per period 
thus: 

.781 198 = present value of i @ 2}^% in 10 periods 
.218802-^.025 = 8.75208 present value of annuity of i 

8.75208X5=43.76040 present value of annuity of 5, the pre- 
mium. 

The method may be stated as follows: 

Compute the interest for i period at the nominal rate on par $30.00 
" I " " " effective " " " 25.00 

Find the difference, which is one rent of an annuity 5.00 

Find the present value of this annuity at the effective rate. 

Another method of computing the cost of a 
bond at a premium is based on the following 
reasoning: 

The purchaser will pay the present value of 
the benefits to be received, which are ordinarily: 

Par at maturity; 
The interest coupons. 

The present value of these benefits will be 
computed at the effective rate. AppHed to the 
preceding illustration, the computation by this 
method would be: 



INTRODUCTION TO ACTUARIAL SCIENCE 47 

Present value of par: 

.781198, present value of i due 10 periods hence at 2}4% 
.781198X1000= 781.198 

Present value of coupons: 

.218802 (compound discount) -r- .025 = 8.75208 

present value of annuity of i 
30.00 (coupons) X 8.75208= 262.562 



Total (as computed above) 1,043.760 

The interest and amortization of premium on 
this bond may be scheduled as follows: 

Amortization — Bond at a Premium 
Cost 1,043.76 

1st interest date: Coupon 30.00 

Interest: 2}4% of 1043.76 26.09 3-91 



2nd 


« 


Carrying value 
" Coupon 

Interest: 2}4% of 1039.85 

Carrying value 
" Coupon 

Interest: 2>^% of 1035.85 

Carrying value 
" Coupon 

Interest: 2}4% of 1031.75 

Carrying value 
" Coupon 

Interest: 2}4% of 1027.54 

Carrying value 
" Coupon 

Interest: 2>^% of 1023.23 


30.00 
26.00 


1,039.85 
4.00 


3rd 


30.00 
25.90 


1,035.85 
4.10 


4th 


30.00 
25.79 

30.00 
25.69 

30.00 
25.58 


1,031.75 
4.21 


Sth 
6th 


1,027.54 

4.31 

1,023.23 

4.42 



Carrying value 1,018.81 



48 INTRODUCTION TO ACTUARIAL SCIENCE 



7th interest date: Coupon 

Interest: 2j^% of 1018.81 

Carrying value 
8th " " Coupon 



30.00 
25.47 



4.53 



9th 



loth 



30.00 
Interest: 2}i% of 1014.28 25.36 

Carrying value 

Coupon 

Interest: 2}4% of 1009.64 



Carrying value 

Coupon 

Interest: 2}47o of 1004.88 

Par — payable at maturity 



30.00 
25.24 

30.00 
25.12 



1,014.28 

4-64 
1,009.64 

476 

1,004.88 

4.88 
1,000.00 



Or the schedule may be shown thus: 





Amortization — Bond . 


AT A Premium 




End of 




Effective 


Premium 


Carrying 


period 


Coupon 


interest 


written off 


value 
1,043.76 


I 


30.00 


26.09 


3.91 


1,039.85 


2 


30.00 


26.00 


. 4.00 


1,035.85 


3 


30.00 


25.90 


4.10 


1,031.75 


4 


30.00 


25.79 


4.21 


1,027.54 


5 


30.00 


25.69 


4.31 


1,023.23 


6 


30.00 


25.58 


4.42 


1,018.81 


7 


30.00 


25.47 


4.53 


1,014.28 


8 


30.00 


25.36 


4.64 


1,009.64 


9 


30.00 


25.24 


4.76 


1,004.88 


10 


30.00 


25.12 


4.88 


1,000.00 



300.00 



256.24 



43.76 



INTRODUCTION TO ACTUARIAL SCIENCE 49 

The second method of computing the price, 
described above, is the better one to use when 
bonds are repayable at a premium. 

Illustration: what price, to net 5%, should be 
paid for a 5^% twenty-year bond of $1,000.00, 
repayable with a bonus of 5%.? 

Present value of 1050 due at maturity: 

.3724306, present value of i at 2}i% in 40 periods 
.3724306X 1050= 391.052 

Present value of coupons: 

.6275694-^.025 = 25.102776 present value of 

annuity of i 
25.102776X27.50= 690.326 

Total 1,081.378 

Computing the Discount 

The two methods described for computing 
the price of a bond sold at a premium may also 
be used when the bond is sold at a discount. 

To illustrate: at what price should a ^1,000.00 
5 year 5% bond be sold to net the investor 6%? 
Interest is payable semi-annually. 

By the first method the difference between the 
interest on par at the effective rate and at the 
nominal rate is computed — the discount is the 
present value of an annuity for the number of 
interest periods, each rent of which is the differ- 
ence in the interest at the two rates. This 



50 INTRODUCTION TO ACTUARIAL SCIENCE 

annuity is discounted at the effective interest 
rate. 

Effective rate: 3% on 1,000= 30.00 

Nominal rate: 2}4% on 1,000= 25.00 

Difference: 5.00 

.744094 is the present value of i at 3% due 10 periods hence: 
•25S9o6-^ .03 = 8.5302 present value of annuity of i. 
8.5302X5 =42.6510, the discount. 
1,000.00— 42.65 = 957.35, the price. 

By the second method the present values of 
par and coupons are computed at the effective 
rate: 

Present value of par: 

.744094X1000= 744-094 

Present value of coupons : 
.255906-r- .03 = 8.5302 
8.5302X25= 213.25s 



Total: 957-349 

The reduction of this discount may be sched- 
uled thus: 

Schedule of Amortization — Bond at Discount 

First period: 

Cost: 957.3s 

Interest: 3% of 957.35 28.72 

Coupon 25.00 3.72 

Carrying value 961.07 

Second period: 

Interest: 3% of 961.07 28.83 

Coupon - 25.00 3.83 

Carrying value 964.90 



INTRODUCTION TO ACTUARIAL SCIENCE 5 1 



Third period: 

Interest: 3% of 964.90 
Coupon 


28.95 

25.00 


3.95 


Carrying value 
Fourth period: 

Interest: 3% of 968.85 
Coupon 


29.07 

25.00 


968.85 
4.07 


Carrying value 
Fifth period: 

Interest: 3% of 972.92 
Coupon 


29-19 

25.00 


972.92 
4.19 


Carrying value 
Sixth period: 

Interest: 3% of 977. 1 1 
Coupon 


29.31 

25.00 

29-44 
25.00 


977.11 
4-31 


Carrjring value 

Seventh period: 

Interest: 3% of 981.42 
Coupon 


981.42 
4.44 


Carrying value 


985.86 


Eighth period: 

Interest: 3% of 985.86 
Coupon 


29.58 

25.00 


4.58 


Carrying value 




990.44 


Ninth period: 

Interest: 3% of 990.44 
Coupon 


29.71 
25.00 


4.71 


Carrying value 




995.15 


Tenth period: 

Interest: 3% 0^995-15 
Coupon 


29.85 
25.00 


4.85 


Carrying value 




lOOO.OQ 



52 INTRODUCTION TO ACTUARIAL SCIENCE 

Or thus : 





Schedule 


OF Amor' 


TIZATION 




eriod 


Effective 
interest 


Coupon 


Discount 


Carr3ring 
value 
957.35 


I 


28.72 


25.00 


3.72 


961.07 


2 


28.83 


25.00 


3.83 


964.90 


3 


28.9s 


25.00 


3.95 


968.85 


4 


29.07 


25.00 


4.07 


972.92 


5 


29.19 


25.00 


4.19 


977.11 


6 


29.31 


25.00 


4.31 


981.42 


7 


29.44 


25.00 


4.44 


985.86 


8 


29.58 


25.00 


4.58 


990.44 


9 


29.71 


25.00 


4.71 


995-15 


10 


29.85 


25.00 


4.85 


1000.00 



Total 292.65 



250.00 



42.65 



Purchases at Intermediate Dates 

When bonds are sold between interest dates, 
the customary method of computing the price is 
as follows: 

Determine the price which would have been 
paid at the last preceding interest date. Deter- 
mine the price which would have been paid at 
the next succeeding interest date. Find the 
difference between these prices. This difference 
is the premium or discount which would be 
amortized for the entire period in which the 
purchase was made. 

Determine the fraction of the period expired 
between the last preceding interest date and the 
date of purchase. 



INTRODUCTION TO ACTUARIAL SCIENCE 53 

Multiply the difference in prices (premium 
or discount for whole period) by the fraction ot 
the period expired. The product is the premium 
or discount for the fractional period. 

Deduct such premium for the fractional 
period from the price at the last preceding interest 
date, or add the discount for the fractional period. 

To the result thus obtained add the accrued 
interest on par at the nominal rate. 

Illustration — bond at premium: what price 
should be paid for a ^loo.oc bond due in 6 years 
and 2 months, bearing 6% and bought on a 5% 
basis plus accrued interest? 

Value 6)/2 years to maturity (per bond table) 
(( /■ « « « « « « 

Difference — premium amortized in 6 months 
Multiply by fraction of period expired — 4 months 

Premium amortized in 4 months 

Price 6}^ years to maturity 
Deduct premium for 4 months 

Value 6 years, 2 months before maturity (flat) 
Interest for 4 months on $100.00 at 6% 

Price including interest 107.25 

At the next interest date the bond will be 
written down to $105.13, its value at that date 
as shown by the bond table. 

Illustration — bond at a discount: what price 
should be paid for a $100.00 bond due in 8 years 



105.49 
105.13 


.36 
% 


105.49 
.24 


105.25 
2.00 



54 INTRODUCTION TO ACTUARIAL SCIENCE 

and I month, bearing 4% and bought on a s3^% 
basis plus accrued interest? 

Value 8 years to maturity (per bond table) 90.40 

" 81^ " " " " " " 89.92 

Difference — discount amortized in 6 months .48 

Multiply by fraction of period expired — 5 months 5/6 

Discount amortized in 5 months .40 

Add value 83^ years before maturity 89.92 

Add accrued interest at 4% on $100 for 5 months 1.67 

Price including interest 91 '99 



Optional Redemption 

When a bond or other obligation gives the 
debtor the option of paying the debt before 
maturity, this right must be taken into consider- 
ation in determining the price to be paid if the 
purchase is to be made at a premium or a dis- 
count. 

If the debtor has the right to redeem at par 
before maturity, the purchase price should be 
computed on the assumption that the right will 
be exercised if the bond is purchased at a pre- 
mium. The reason for the assumption can be 
shown by a comparison of prices in a bond table. 

A 6% bond payable in 20 years bought on a 
5% basis should cost 11^.55 

A 6% bond payable in 15 years bought on a 
5% basis should cost 110.47. 



INTRODUCTION TO ACTUARIAL SCIENCE 55 

Now if the bond is payable in twenty years 
with an option to redeem in fifteen years the 
purchaser may be buying a bond with only fif- 
teen years to run; and he should pay for it on 
the supposition that it will be paid at the optional 
date. 

On the other hand if the bond is to be pur- 
chased at a discount, he should assume that it 
will not be paid until maturity. A bond table 
shows that a 5% bond payable in 15 years 
bought on a 6% basis should cost 90.20; a 5% 
bond payable in 20 years bought on a 6% basis 
should cost 88.44. 

If the purchaser pays 90.20 for the bond and 
it is not paid for twenty years, he will not earn 
6% on his investment. In fact, he will earn a 
little less than S%%. 

If the debtor must pay a premium in order to 
redeem the bond before maturity, the purchaser 
should assume that the option will not be exer- 
cised in case the bond is to be sold at a discount. 
It was shown in the preceding paragraph that a 
5% bond sold on a 6% basis would sell for ^90.20 
if redeemable at par at the end of 15 years. If 
redeemable at a premium, it would sell for a still 
higher price. But a purchaser would be unwise 
to pay this higher price when there is a possi- 
bility that the bond will run the full twenty, 
years. He should buy on the assumption that 



56 INTRODUCTION TO ACTUARIAL SCIENCE 

the option will not be exercised. If it is exercised 
his rate of earning will be more than 6%. 

But if the bond is to be purchased at a pre- 
mium, and if the debtor must pay a premium to 
redeem the bond before maturity, the purchaser 
cannot assume that the option will be exercised, 
nor can he assume that it will not be exercised. 
The advantage to the debtor arising from the 
payment at par on an optional maturity date 
may vanish if he has to pay a premium if he 
redeems before maturity. Whether or not it 
will be advantageous will depend on the amount 
of the premium. Therefore, the purchaser should 
compute the price to be paid on the given basis 
if the bond runs to maturity, and the price to 
be paid if the option to redeem at a premium is 
exercised; and he should then pay the lower 
price. 

To illustrate: on a 5% basis what should be 
paid for a ^1,000.00 6% bond, due in 20 years, 
with a privilege of redemption in 15 years at no? 

A bond table shows the value on a 5% basis 
of a 20 year 6% bond to be ^1,125.50. 

The value if the option is exercised could be 
computed thus: 

Value of 1,100 in 15 years: 
Present value of i at 23^% due 30 periods hence. .476742685 
Multiply by iioo 

524.4169535 



INTRODUCTION TO ACTUARIAL SCIENCE 57 

Value of coupons: 

Present value of i, as above .476742685 

Compound discount -523257315 

.5 23 25 73 1 5 -T". 023^2 = 20.93 02926, present value of annuity 
of I 
20.9302926X 30 = 627.908778 
Value of par and premium 524.4169 

Value of coupons 627.9087 

Total 1 152.3256 

The price, on the assumption that the bond 
runs to maturity, is ^1,125.50. On the assump- 
tion that it is paid at the optional date, the price 
is ^1,152.33. The purchaser should assume that 
the option will not be exercised and pay ^1,125.50. 

If the premium to be paid at the optional 
redemption date is not too large, it may still be 
desirable to exercise the option. 

To illustrate: what price should be paid for 
the bond in the preceding illustration if the 
optional redemption price is loi instead of no? 
Price if option is not exercised: $1,125.50, as 
above. 
Price if option is exercised : 

Present value of loio in 15 years: 

Present value of i at 2j^% due 30 periods hence .476742685 
Multiply by loio 

481.5101 
Present value of coupons — as above 627.9087 

Total 1,109.4188 

This price, $1,109.42, should be paid because it is less than 
$1,125.50. 



58 introduction to actuarial science 

Computing the Rate on Bonds Sold at 
Premium or Discount 

When a bond is purchased on the basis of an 
effective rate other than the nominal rate, it is 
a simple matter to compute the price to be paid; 
but when the bond is purchased at a price not 
listed in the bond tables, it is by no means an 
easy matter to compute the price. In fact, there 
is no mathematical formula which can be applied 
to determine the rate exactly. The rate can be 
approximated in several ways, two of which will 
be explained. 

A 25 year 6% ^100 bond, interest payable 
semi-annually, is purchased for ^110.38. What 
is the effective rate? 

This price of ^110.38 is shown by a bond table 
to be the price on a 5.25% basis. The bond table 
shows : 





25 years 6% 




5.20% 




III. 12 


S.25 




110.38 


S.30 




109.641 



But let us assume that the 110.38 is not 

shown by the table and that the nearest values 
are 

On a 5.20 basis 111.12 

On a 5.30 " 109.64 



INTRODUCTION TO ACTUARIAL SCIENCE 59 

Then .10 of one per cent, difference in the 
rate causes 1.48 difference in price. 

Price on a 5.20 basis 111.12 

Price given in illustration 110.38 

Difference .74 

Then to find the approximate rate, add to 5.20% 

^3 of .10 of 1% 

^of.io = .os 

5.20% + .05% = 5.25 % the rate. 
This rate happens to be exactly correct, but 
it is very unusual to obtain exact results by inter- 
polation. 

The rate may be approximated by the follow- 
ing formulas when bond tables are not available 
for interpolation: 

Bond at a premium: 

2(1 — Pr) 

'■.(CH-P + ?) 

Bond at a discount: 

_ 2 (I + D) 



.(c+P-f) 



The symbols used are: 

r — effective rate per period 

P— par 

/—total interest on bonds 

Pr — premium 

D — discount 

C — cost (par + premium; or par — discount). 

w— number of interest periods. 



6o INTRODUCTION TO ACTUARIAL SCIENCE 

Applying the first formula to the illustration : 

2 (150 — 10.38) ^ ^ . J 

r = — ^^ ^Q gv = 2.65% per period 

SO r 110.38 + 100 + -7^ ) 

or 5-30% per annum. 

It will be noted that this result is much less 
exact than the one obtained by interpolation — 
still it is useful when one has no bond table and 
desires to obtain a rough approximation of the 
rate. 

Illustration of bond at a discount: a 10 
year 5% ^100 bond is bought at 96.94. Interest 
payable semi-annually. What is the approxi- 
mate effective rate ^ 

2 (50 + 3.06) ^ ^ .J 

r = — y ^^ ^ — To6\^ 2.692% per period 

20 ^96.94 + 100 " ^ j 

or 5.384% per annum. 

The true effective rate is 5.40%. 

This method produces a rate which is too 
large on bonds sold at a premium and too small 
on bonds sold at a discount. The error is due to 
the fact that the formula is based on arithmetical 
progression, while the amortized premium or 
discount increases or decreases periodically in only 
an approximate arithmetical progression. 



introduction to actuarial science 6l 

Depreciation 

Two depreciation methods — the annuity and 
sinking fund methods — involve compound in- 
terest. When the annuity method is used, the 
investment in the depreciating asset is dealt with 
as if it were an investment in an annuity. The 
periodical depreciation charges are analogous to 
rents and must be large enough to exhaust the 
cost of the asset, or the cost less residual value, 
and also provide for the interest. In other words, 
the charge to operations for depreciation must 
provide for credits to interest for interest on 
the gradually diminishing investment, and for 
credits to the depreciation reserve. The amount 
of the credit to interest is coniputed by multi- 
plying the carrying value of the asset (cost less 
reserve at beginning of period) by the interest 
rate. The credit to the reserve is the difference 
between the charge to depreciation and the 
credit to interest. 

When there is no scrap value, the formula for 
computing the periodical depreciation, is 

In this formula: 

d = periodical depreciation 

c = cost of asset 

(^1 = present value of annuity of i 



62 INTRODUCTION TO ACTUARIAL SCIENCE 

Illustration: what is the annual depreciation 
on an asset costing ^5,000 which will have no 
value at the end of five years, if depreciation is 
to be computed by the annuity method using a 
rate of 5%? Present value at 5% of i due 5 
periods hence is .783526166. 

Then .216473834-7- .05 =4.32947668, or a-[ 

5000 _ ^ ^^^ o^ 

a = 777- = 1,154.07. 

4.32947668 > ^^ ^ 

The annual depreciation entries may be tabu- 
lated thus: 





Debit 


Credit 


Credit 


Carrying 


fear 


depreciation 


interest 


reserve 


value 
5,000.00 


I 


1,154.87 


250.00 


904.87 


4,095.13 


2 


1,154.87 


204.76 


950.11 


3,145.02 


3 


1,154.87 


157.25 


997.62 


2,147.40 


4 


1,154.87 


107.37 


1,047.50 


1,099.90 


5 


1,154.87 


55.00 


1,099.87 


.03 



When there is a scrap value, the formula is : 

r-(/XP) 



d = 



a-. 
»i 



In which the symbol s represents scrap-value 
and P represents the present value of i. 

Assuming that the asset in the preceding 
illustration will have a residual value of ^2,000 
at the end of 5 years, what should be the annual 
depreciation .r^ 



INTRODUCTION TO ACTUARIAL SCIENCE 63 

The present value of i at 5% due in 5 periods, was stated in 
the illustration to be .783526166; and the present value of an 
annuity of i for 5 periods at 5% was computed above, as 
4.32947668. Then 

, 5000— (2000 X .783526166) 

' = i:ii^ — = 79-9^. 

The annual depreciation entries may be tabu- 
lated thus: 





Debit 


Credit 


Credit 


Carrying 


^ear 


depreciation 


interest 


reserve 


value 
5,000.00 


I 


792.92 


250.00 


542.92 


4,457.08 


2 


792.92 


222.85 


570.07 


3,887.01 


3 


792.92 


194-35 


598.57 


3,288.44 


4 


792.92 


164.42 


628.50 


2,659.94 


S 


792.92 


133.00 


659.92 


2,000.02 



The sinking fund method is based on the 
assumption that a fund is created at compound 
interest to equal the total depreciation. If a 
fund is created, the contribution is computed in 
accordance with the formula already stated and 
explamed, namely: 

S.F.C. = S.F. - J-, 

Since the total fund required is the difference 
between the cost and the scrap value of the asset, 
the formula for determining the periodical con- 
tribution to the fund is 

S.F.C.= ^-^^ 

•^51 



64 INTRODUCTION TO ACTUARIAL SCIEKCE 

Illustration: what annual contribution should 
be made to a fund on a 5% basis, compounded 
annually, to provide for an asset costing ^5,000 
and expected to have a residual value of ^2,000 
at the expiration of 5 years ? And what should be 
the annual entries for the fund and the deprecia- 
tion reserve ? i .05^ = i .276282. 

j^ =.276282-7- .05 =5.52564 





s.F.c.=5^^^-^;^^= 
5.52564 


= 542.92 






Table of Fund Entries 






Debit Credit 


Credit 


Balance 


^ear 


fund interest 


cash 


fund 


I 


542.92 


542.92 


542.92 


2 


570.07 27.15 


542.92 


1,112.99 


3 


598.57 55.65 


542.92 


1,711.56 


4 


628.50 85.58 


542.92 


2,340.06 


S 


659.92 117.00 


542.92 


2,999.98 



The reserve should keep pace with the fund 
so that at the end of the anticipated life of the 
asset, the fund and the reserve each will equal 
the total depreciation. Therefore the amount 
charged each year to the fund, as shown in the 
''debit fund" column, should be charged to 
depreciation and credited to the reserve for depre- 
ciation. 



SOLUTIONS TO ACTUARIAL PROBLEMS 

IN THE AMERICAN INSTITUTE 

EXAMINATIONS 

As this book is designed to assist those who 
expect to be candidates in the examinations set 
by the American Institute of Accountants, thtj 
following problems and solutions are given^ 
These are all the problems in actuarial science 
given in the Institute examinations at the time 
of publishing this book. 

Problem i 
(June 1917) 

A machine costing ^81.00 is estimated to 
have a life of four years, with a residual value of 
^16.00. Prepare a statement showing the annual 
charge for depreciation according to each of the 
following methods: 

(a) Straight line. 

(b) Constant percentage of diminishing 

value. 

(c) Annuity method. 

(For convenience in arithmetical calculation 

assume the rate of interest to be io%.) 

65 



(ig introduction to actuarial science 
Solution Problem i 

(a) The straight-line method does not require 
the appHcation of actuarial science, the annual 
depreciation being computed as follows: 

81.00—16.00 



Year 

I 

2 
3 
4 

Total 65.00 

(b) The constant percentage of diminishing 
value method requires the following computation 

to determine the rate. 

\6 



\ 


■ \.\j,\i.^ 




statement 


follows 


\ 


Depreciation 


Carrying Value 






81.00 


16.25 




647s 


16.25 




48.50 


16.25 




32.25 


16.25 




16.00 



Rate = 



V 81 



Fortunately for the person who is required to 
solve this problem, both the numerator and 
denominator of the fraction \\ are fourth powers 
of integers. 16 is the fourth power of 2, and 81 
is the fourth power of 3. 



Hence 



4/16 
V8l== 



and r = I — ^ 



'=-1^ 



INTRODUCTION TO ACTUARIAL SCIENCE 67 

Usually it is necessary to employ logarithms 
to determine the rate to be applied to the 
diminishing value. The computation by loga- 
rithms is therefore shown although the applicant 
in an examination could not be expected to sub- 
mit a logarithmic solution. 

81 
Log 16 = 1.204120 or 1 1. 204120— 10 
Log 81 = 1.908485 

Then 11. 204120— 10 

Minus 1.908485 

9-295635 -10 = Log 16/81 

Extract the 4th root: 

9-295635-10 
Add 30 —30 

39.295635-40 

Divide by 4: 

39-295635-40 ^ 9.823909- 10, the log o(\/l6/Si 
4 

9.823909 — 10 is the log of .6666+ 

Then r = I — .6666 = .3333 

The required table follows: 

Year Depreciation Carrying value 

81.00 

1 (J/^ of 81.00) 27.00 54.00 

2 (3^ of 54.00) 18.00 36.00 

3 (H of 36.00) 12.00 24.00 

4 04 ^f 24.00) 8.00 16.00 

65.00 



68 INTRODUCTION TO ACTUARIAL SCIENCE 

(c) The annual depreciation according to the 
annuity method is computed by the formula 

In this formula 

c = cost =^8i.oo 

s= scrap value =$16.00 

P = present value of i for 4 periods at 10% 
1.10^=1.4641 
.4641-r- 1.4641 = .3 1699 compound discount 
and I— .3 1699 = .68301 present value 

Then d= 8i-(f3°ixi6) 



year 



.^•31699-^.10 

_ 81 — 10.92816 

3.1699 






= 22.11 






equired table follows: 




Interest 
credited 


Depreciation 
debited 


Carrying 
value 
81.00 


8.10 


22.11 


66.99 


6.70 
S.16 
3.46 


22.11 
22.11 
22.11 


51.58 
34-63 
15.98 



Total 23.42 88.44 

Problem 2 

(June 1917) 

Argument has been strongly urged that aside 
from any question of possible mismanagement, 
or of the difficulty of making satisfactory invest- 
ments to yield the same rate as is paid on the 



INTRODUCTION TO ACTUARIAL SCIENCE 69 

bonds, a sinking fund for bonds is more expensive 
than an arrangement for the serial repayment of 
bonds. Tbis is illustrated by the. case of ^20,000 
5% bonds.- If these are paid off in a series, 
one each year, the total payment made will be 
principal ^20,000, interest ^10,500, total ^30,500. 
The annual sinking fund to pay these bonds 
would on a 5% basis amount to $604.85, making 
in twenty years $12,097, ^^d the interest paid 
on the bonds would be $20,000, total payments 
$32,097. The apparent excess burden is accord- 
ingly $1,597. 

Discuss the above argument and show clearly 
just what the figures mean and in what the 
apparent saving actually consists. 

Solution Problem 2 

The disadvantage of the sinking fund plan is 
apparent only, and not real. The apparent excess 
burden of $1,597 is due to the fact that during 
the early years more principal is provided for 
under the serial redemption plan than under the 
sinking fund plan. Since the borrower has more 
use of the money under the sinking fund plan 
than under the serial redemption plan, the 
necessity of paying more interest under this plan 
cannot be said to be a disadvantage. 

The relative advantages of these two plans 
may be made more apparent by a brief discussion 



70 INTRODUCTION TO ACTUARIAL SCIENCE 

of a third plan, known as the annuity method. 
By this method equal annual payments would be 
made, these payments including the interest 
accrued to date and a payment on the principal. 
The annual payments under the annuity plan 
would be computed as follows: 

The ^20,000 principal is the present value of 
the 20 payments or rents of unknown amount 
which will pay the principal sum and the interest. 

.376889 is the present value of i at 5% due 20 periods hence 

.623111 is the compound discount 

.61311 1 -r- .05 = 12,4622 the present value of an annuity of 

I for 20 periods 
20,000 -T- 12.4622 =^ 1604.85 the equal annual payment to be 

made under the annuity plan. 

At the end of the first year there will be 
^1,000.00 accrued interest; the remaining ^604.85 
will apply on the principal. The operation of 
the three plans during the first year may be com- 
pared thus: 

Serial Sinking Annuity 

redemption fund redemption 

Interest 1,000.00 1,000.00 1,000.00 

Payment on principal 1,000.00 604.85 604.85 

Remaining principal 19,000.00 19,395- 15 I9>39S-IS 

Since the principal unprovided for during the 
second year is greater under the sinking fund and 



INTRODUCTION TO ACTUARIAL SCIENCE 7 1 

the annuity plans than under the serial plan, it is 
reasonable to expect the interest expense of the 
second year to be larger under these plans. The 
interest on these excesses during the entire life 
of the loan exactly accounts for the ^1,597.00, 
as shown by the following comparative table: 

Serial Redemption Plan 



Year 


Total 


Payment 


Payment 


Balance of 




disbursement 


of interest 


on principal 


principal 
20,000.00 


I 


2,000.00 


1,000.00 


1,000.00 


19,000.00 


2 


1,950.00 


950.00 


1,000.00 


18,000.00 


3 


1,900.00 


900.00 


1,000.00 


17,000.00 


4 


1,850.00 


850.00 


1,000.00 


16,000.00 


5 


1,800.00 


800.00 


1,000.00 


15,000.00 


6 


1,750.00 


750.00 


1,000.00 


14,000.00 


7 


1,700.00 


700.00 


1,000.00 


13,000.00 


8 


1,650.00 


650.00 


1,000.00 


12,000.00 


9 


1,600.00 


600.00 


1,000.00 


11,000.00 


10 


1,550.00 


550.00 


1,000.00 


10,000.00 


II 


1,500.00 


500.00 


1,000.00 


9,000.00 


12 


1,450.00 


450.00 


1,000.00 


8,000.00 


13 


1,400.00 


400.00 


1,000.00 


7,000.00 


14 


1,350.00 


350.00 


1,000.00 


6,000.00 


IS 


1,300.00 


300.00 


1,000.00 


5,000.00 


16 


1,250.00 


250.00 


1,000.00 


4,000.00 


17 


1,200.00 


200.00 


1,000.00 


3,000.00 


18 


1,150.00 


150.00 


1,000.00 


2,000.00 


19 


1,100.00 


100.00 


1,000.00 


1,000.00 


20 


1,050.00 


50.00 


1,000.00 


0.00 ' 




30,500.00 


10,500.00 


20,000.00 





72 INTRODUCTION TO ACTUARIAL SCIENCE 

Sinking Fund and Annuity Plans 



Year 


Total 


Payment 


Payment 


Balance 




disbursement 


of interest 


on principal 


principal 
20,000.00 


I 


1,604.85 


,000.00 


604.85 


19,395.15 


2 


1,604.85 


969.76 


635.09 


18,760.06 


3 


1,604.85 


938.00 


666.85 


18,093.21 


4 


1,604.85 


904.66 


700.19 


17,393.02 


5 


1,604.85 


869.65 


735.20 


16,657.82 


6 


1,604.85 


832.89 


771.96 


15,885.86 


7 


1,604.85 


794.29 


810.56 


15,075.30 


8 


1,604.85 


753.77 


851.08 


14,224.22 


9 


1,604.85 


711. 21 


893.64 


13,330.58 


10 


1,604.85 


666.53 


938.32 


12,392.26 


II 


1,604.85 


619.61 


985.24 


1 1,407.02 


12 


1,604.85 


570.35 


1,034.50 


10,372.52 


13 


1,604.85 


518.63 


1,086.22 


9,286.30 


14 


1,604.85 


464.32 


1,140.53 


8,145.77 


IS 


1,604.85 


407.29 


1,197.56 


6,948.21 


i6 


1,604.85 


347.41 


1,257.44 


5,690.77 


17 


1,604.85 


284.54 


1,320.31 


4,370.46 


i8 


1,604.85 


218.52 


1,386.33 


2,984.13 


19 


1,604.85 


149.21 


1,455.64 


1,528.49 


20 


1,604.85 


76.42 


1,528.43 


.06 




32,097.00 


12,097.06 


19,999.94 





By comparing the two "Balance of Principar' 
columns it will be seen that after the first year, 
the principal is always larger under the sinking 
fund and annuity plans than under the serial 
redemption plan. Therefore the interest under 
the sinking fund plan is larger than under the 



INTRODUCTION TO ACTUARIAL SCIENCE 73 

serial plan. But so long as the interest is propor- 
tionate to the principal, this cannot be said to be 
a disadvantage. 

Problem 3 
(November 1917) 

You are called upon to state what is the 
annual sinking fund necessary to redeem a 
principal sum of ^1,000,000 due 30 years hence — 
it being assumed that the annual sums set aside 
are invested at compound interest at 5 per cent. 
State what computations you would make to 
arrive at the result desired. You need not work 
out the computation. 

Solution Problem 3 

Assuming that the contributions are deposited 
at the end of each year, the annual sum would be 
computed by dividing the required fund of 
$1,000,000 by the amount of an ordinary annuity 
of $1 for 30 periods at 5%. 

The amount of an ordinary annuity of $1.00 
for 30 periods at 5% would be computed by divid- 
ing the compound interest on $1.00 for 30 periods 
at 5% by the simple interest rate 5%. 

The compound interest would be computed 
by raising 1.05 to the 30th power, and deducting 
I therefrom. 



74 INTRODUCTION TO ACTUARIAL SCIENCE 

Or, Stated in the order of procedure, the 
annual contribution would be computed thus: 

Raise 1.05 to the 30th power, thus: 
1.05 X1.05 =1.052 
1.052 X 1.052 = 1.054 

1.054 Xi.o54=i.o5S 

1.055 X 1.05^=1.05^6 
1.0516X1.058=1.0524 
1.0524x1.054=1.0528 
1.0528X1.052 = 1.0530 

Compute the compound interest, /, which is 1.053°— i 

Divide the value of / by .05 to compute the amount of an 

annuity of i, represented by s^. 
Divide $1,000,000 byj"^ to determine the annual contribution. 

Problem 4 

(November 1917) 

A owns an annuity of ^50 per annum, the 
first payment on which falls due one year hence, 
and which continues for a period of twenty years 
certain. State: 

(a) The present value of the benefit 

(b) The amount which he will have accumu- 

lated at the end of the period if he invests 
each moiety as it becomes due. 

Assume interest at 4 per cent payable an- 
nually. In this connection the value of (1.04)20 
is stated to be equal to 2. 191 123. 



introduction to actuarial science 75 

Solution Problem 4 

(a) The present value of the benefit or annuity 
is computed by dividing the compound 
discount by the simple interest rate of 4%. 
Since the compound discount is not stated 
in the problem it must be computed. Since 
the problem states the amount of i at 4% 
for 20 periods the easiest way to compute 
the compound discount is to divide the 
compound interest on i by the amount of i. 

2.191 123 = amount of i for 20 periods at 4% 

1. 000000 



1. 19 1 1 23 = compound interest on i for 20 periods at 4% 



And 1.191123^2.191123 = .543613, compound discount 

Then .543613 -5- .04 = 13.590325, present value of annuity of i 

And 13.590325X50=679.52, present value of annuity of 50 

(b) The amount which will have accumulated at 
the end of the period if each moiety or rent 
is invested as it becomes due, is the amount 
of an ordinary annuity of ^50.00 per 
period for 20 periods invested at 4%. The 
amount of an annuity of ^i is computed 
by dividing the compound interest on i at 
4% for 20 periods by the simple interest 
rate of 4%; the amount of an annuity of 
I should then be multiplied by 50. 



76 INTRODUCTION TO ACTUARIAL SCIENCE 

2. 19 1 123 amount of i at 4% for 20 periods 

i.oooooo 



1. 191 123 compound interest on i at 4% for 20 periods. 



1.191 123 -^ .04 = 29.778075, amount of annuity of i 
29.778075X50=1488.90 " " " " so 

Proof of Present Value of ^679.52 



Period 


Rent 


Interest 


Reduction 


Balanc 
679.52 


I 


50.00 


27.18 


22.82 


656.70 


2 


50.00 


26.27 


23.73 


632.97 


3 


50.00 


25.32 


24.68 


608.29 


4 


50.00 


24.33 


25.67 


582.62 


5 


50.00 


23.30 


26.70 


555.92 


6 


50.00 


2Z0Z4 


27.76 


528.16 


7 


50.00 


21.13 


28.87 


499.29 


a 


50.00 


19.97 


30.03 


469.26 


9 


50.00 


18.77 


31.23 


438.03 


10 


50.00 


17.52 


32.48 


405.55 


II 


50.00 


16.22 


33.78 


371.77 


12 


50.00 


14.87 


35.13 


336.64 


13 


50.00 


13.47 


36.53 


300.11 


14 


50.00 


12.00 


38.00 


262.11 


IS 


50.00 


10.48 


39.52 


222.59 


16 


50.00 


8.90 


41.10 


181.49 


17 


50.00 


7.26 


42.74 


138.75 


18 


50.00 


5-55 


44.45 


94.30 


19 


50.00 


3.78 


46.22 


48.08 


20 


50.00 


1.92 


48.08 


.00 


Totals 


1,000.00 


320.48 


679.52 





INTRODUCTION TO ACTUARIAL SCIENCE 77 



Proof of Amount of ^1,488.90 
Period Interest Rent 



Amount 



I 




50.00 


50.00 


2 


2.00 


50.00 


102.00 


3 


4.08 


50.00 


156.08 


4 


6.24 


50.00 


212.32 


S 


8.49 


50.00 


270.81 


6 


10.83 


50.00 


331.64 


7 


13.27 


50.00 


394.91 


8 


15.80 


50.00 


460.71 


9 


18.43 


50.00 


529.14 


10 


21.17 


50.00 


600.31 


II 


24.01 


50.00 


674.32 


12 


26.97 


50.00 


751.29 


13 


30.05 


50.00 


831.34 


14 


33.25 


50.00 


914.59 


15 


36.58 


50.00 


1,001.17 


16 


40.05 


50.00 . 


1,091.22 


17 


43.65 


50.00 


1,184.87 


18 


47.40 


50.00 


1,282.27 


19 


51.29 


50.00 


1,383.56 


20 


55-34 


50.00 


1,488.90 


Totals 


488.90 


1000.00 






Problem s 






(May 


1918) 





In auditing the books of a corporation you 
find that, in order to provide a sum to redeem a 
mortgage of ^100,000.00 faUing due at the end of 
10 years, a reserve of ^8,000.00 per annum has 
been set aside annually for three years, but that, 



78 INTRODUCTION TO ACTUARIAL SCIENCE 

contrary to intention, the company has failed to 
accumulate interest thereon. Assuming interest 
at 4 per cent, (convertible annually) what should 
have been the total accumulations to date, and 
what amount should now be set aside annually 
for the next seven years in order to complete the 
sinking fund? (1-04)^=1.31593. 

Solution Problem 5 

This problem may be construed in several 
ways. The first uncertainty of meaning is caused 
by using the word ^'reserve.'' A reserve is an 
account with a credit balance set up, usually, by 
a charge to profit and loss or to surplus. 

Under a strict definition of the word "re- 
serve,'' the problem means that an entry has 
been made for the appropriation of surplus by 
charging surplus and crediting a reserve, and 
that no fund has been set aside, since none is 
mentioned. The examiners, however, may have 
used the word "reserve'' to convey the idea that 
a fund has been set aside. We have two possible 
conditions, therefore: 

(a) A reserve of ^24,000, but no fund 

(b) A fund of ^24,000. 

The following question is also uncertain in 
meaning: "Assuming interest at 4 per cent 



INTRODUCTION TO ACTUARIAL SCIENCE 79 

(convertible annually) what should have been the 
total accumulation to date?" Do the examiners 
want to know 

1. How much the three ^8,000 deposits would 

have amounted to if invested at 4%? 
If so, the solution is : 

1.04^ = 1. 1 24864 amount of i at 4% for 3 periods 

.124864-7- .04 = 3.1216 amount of annuity of i at 4% for 3 

periods 
3.1216X8,000. = 24,972.80 amount which would be on hand. 

2. Or, do they wish to know how much should 

have been on hand if the corporation had 
computed the annual contributions to 
the fund on a strict actuarial basis, and 
had accumulated interest thereon.? If 
so, the solution is: 

1.04' =1.31593 (per problem) 

1.04^ =1.124864 (as above) 

1.04^° = 1.04^X1.043 = 1.3 1593X1.124864= 1.480242 

.480242 -7- .04 = 12.00605 amount of annuity of i for 10 periods 

at 4% 
ioo,ooo-M 2.00605 = 8,329.13 annual contribution 
Amount of annuity of i for 3 periods, as above = 3.i2i6 
8,329.13X3.1216 = 26,000.21 amount which would be on hand. 

The second part of the question is: ^^What 
amount should now be set aside annually for 
the next seven years in order to complete the 
sinking fund?" 



8o INTRODUCTION TO ACTUARIAL SCIENCE 

(a) If there is a reserve of ^24,000 but no fund 

at the date of the audit, the entire 
$100,000 will have to be provided by 
the seven contributions and interest 
thereon. 

1.047=1.31593 

.31593^.04 = 7.89825 amount of annuity of i at 4% 

for 7 periods 
100,000^7.89825 = 12,661.03 periodical contribution 

(b) If there is already a fund of $24,000 this 

fund will draw interest for the remain- 
der of the 10 years. Since the first three 
contributions would normally have 
been made at the end of the first three 
years, the remaining seven payments 
would begin at the end of the fourth 
year. This would mean that although 
seven more deposits are to be made 
there are only six years until the matur- 
ity of the mortgage. 

I '3 1 593 "^1-04 = 1-2653 2, value of 1.04^ 

1.26532X24,000 = 30,367.68 amount to which 24,000 will 

accumulate. 
Total fund required 100,000.00 

24,000.00 and interest thereon 30,367.68 

Balance to be provided by 7 installments 69,632.32 



69,632.32-^7.89825 = 8,816.18 annual contribution. 



introduction to actuarial science 8 1 

Problem 6 

(November 1918) 

A corporation wants to retire a debt of 
^105,000 bearing 5% interest payable annually. 
The tenth payment, including interest, is to be 
^15,000. The other nine periodical payments are 
all to include interest and to be of the same 
amount. Required the amount of each of such 
nine payments. (1.05^ = 1.551328.) 

Solution Problem 6 

The words ^^ retire a debt'' indicate that the 
payments are to be made to the creditor and not 
into a sinking £und. Since the last payment of 
$15,000 covers the balance of the principal 
unpaid at the beginning of the tenth year with 
5% interest thereon, this $15,000 payment must 
be 1.05 times the balance unpaid at the beginning 
of the tenth year. 

Hence, $i5,ooo.oo-M%05 =$14,285.71 balance unpaid at beginning 

of lothyear. 
$105,000.00 total debt. 

14,285.71 principal liquidated by loth payment] 

$90,714.29 " " " first 9 payments 

Then ^90,714.29 is the present value at 5% of 
nine unknown annual payments, which are com- 
puted thus: 



82 INTRODUCTION TO ACTUARIAL SCIENCE 

1.05^=1.551328 (per problem) 
Then .551328 (comp. int.) -M.55 1328 (amt. of i) = .355391 com" 
pound discount on i for 9 periods at 5% 
•355391"^ 'OS = 7* 107^2 present value of annuity of i for g 
periods at 5%, or principal which would be paid by 9 
annual installments of $1.00 
90,7I4.29-^ 7. 10782 = 12,762.60 number of dollars in each of 
the first 9 annual payments made in liquidation of 
$90,714.29 of principal, and interest thereon. 

In addition to the annual payments of ^12,- 
762.60 which Hquidate ^90,714.29 and the inter- 
est on the diminishing balance thereof during the 
first nine years, the corporation must pay during 
each of these years 5% interest on the ^14,285.71 
unpaid at the beginning of the tenth year. 5% 
of ^14,285.71 is ^714.29. 







SUMMARY 




$12,762 


.60 annual payment during 


first 9 years on 


$90,714.29 


714.29 annual i 


nterest " 


<C (( « (( 


14,285.71 


$13,476.89 total 












PROOF 










Annual Payment 




Year 


Total 


Interest 


Principal 


Balance 
105,000.00 


I 


13,476.89 


5,250.00 


8,226.89 


96,773.11 


^ 


13,476.89 


4,838.66 


8,638.23 


88,134.88 


3 


13,476.89 


4,406.74 


9,070.15 


79,064.73 


4 


13,476.89 


3,953.24 


9,523.65 


69,541.08 


5 


13,476.89 


3,477.05 


9,999.84 


59,541.24 


6 


13,476.89 


2,977.06 


10,499.83 


49,041.41 


7 


13,476.89 


2,452.07 


11,024.82 


38,016.59 


8 


13,476.89 


1,900.83 


11,576.06 


26,440.53 


9 


13,476.89 


1,322.03 


12,154.86 


14,285.67 



Totals 136,292.01 


31,291.96 


105,000.05 



INTRODUCTION TO ACTUARIAL SCIENCE 83 

There is an over payment of five cents, a dis- 
crepancy too small to be avoided. 

Problem 7 

(November 1918) 

A $10,000 five per cent semi-annual coupon 
bond is bought on a 4 per cent basis, due i}/2 
years hence. What did it cost? 

Solution Problem 7 

Since the bond is due in ij^ years and bears 
semi-annual coupons, there are 3 interest periods 
and the effective rate is 2% per period. The cost 
of the bond may be computed in two different 
ways, but in either case the present value of an 
annuity of i at 2% for 3 periods will be required, 
and it is therefore computed thus: 

i.oooooo-M.02 = .98o392 P.V. of i at 2% due i period hence 
.98o392-M.02 = .96ii69 " " i "2% " 2 periods " 
.961 169-- 1.02 = .942322 " " I "2% "3 " •• 

I — .9423 22 = .057678 compound discount 

.057678 -j- .02 = 2.8839 P- V. of annuity of i at 2% for 3 periods 

First Method : 

Find the present value of all benefits under 
the bond, discounted at 2%: 

Par (10,000.00) 

.9423 22 (P. V. of I due 3 periods hence) X 10,000 = 9,423 .22 
Coupons (250.00 each) 

2.8839 (P. V. of annuity of i for 3 periods) X 250= 720.98 

10,144.20 



84 IKTRODUCTION TO ACTUARIAL SCIENCE 

Second Method: 

Find the present value discounted at the 
•effective interest rate, of an annuity whose rents 
are equal to the difference between the interest 
at the nominal rate and at the effective rate on 
par. The present value of this annuity is the 
premium. 

2j^% of $10,000=^250.00 nominal interest on par 
2% " 10,000= 200.00 effective " " " 

50.OD difference 

50.00X2.8839 (P. V. of Annuity) = 144.195 premium 
10,000.00 par 
144.20 premium 





10,144.20 cost 


of bond. 
PROOF 












jric 


>d Coupon 


Effective 


Premium 


Carrying 






mterest 


amortized 


value 
10,144.20 


I 


250.00 


202.88 


47.12 


10,097.08 


2 


250.00 


201.94 


48.06 


10,049.02 


3 


250.00 


200.98 


49.02 


I0;000.00 




750.00 


605.80 


144-20 





Problem 8 

(May 1919) 

A bond, bearing interest at 5% per annum 
payable annually, and repayable in five years, 
with bonus of 10%, is for sale. What price can a 



INTRODUCTION TO ACTUARIAL SCIENCE 85 

purchaser pay who desires to reaHze 6% on his 
investment ? (V^ at 6% = .7473) 

Solution Problem 8 

When a bond is repayable at a premium, the 
best method of computing the price is to find the 
present value of the benefits to be received, thus: 

Par plus 10% bonus: 

Assuming a par of ^1,000.00 the payment at 
maturity would be ?i, 100.00 

Then 1,100X7473 (P.V. of I in 5 periods at 6%) = 822.03 
Coupons : 

I — .7473 =.2527 compound discount 

.2527-j-.o6=4.2ii6 P. V. of annuity of i 

50.00 (Coupons) X4.21 16= 210.58 





Total price 


PROOF 




1,032.61 


^ear 


Coupon 


Effective 
income 


Amortization 


Carrying 

value 
1,032.61 


I 


50.00 


61.96 


11.96 


1.044.57 


2 


50.00 


62.67 


12.67 


1,057.24 


3 


50.00 


63.43 


13.43 


1,070.67 


4 


50.00 


64.24 


14.24 


1,084.91 


5 


50.00 


65.09 


15.09 


1,100.00 




250.00 


317.39 


67.39 





Problem 9 

(May 1 91 9) 

A lease has five years to run at ^1,000.00 a 
year payable at the end of each year, with an 
extension for a further five years at ^1,200.00 a 



86 INTRODUCTION TO ACTUARIAL SCIENCE 

year. On a 6% basis what sum should be paid 
now in lieu of the ten years' rent ? (V^ at 6% = 

•7473) 

Solution Problem 9 

The sum to be paid for the first 5 rents of 
^1,000.00 each is the present value of an ordinary 
annuity, computed as follows: 

•7473 present value of i at 6% for 5 periods 

I — .7473 =.2527 compound discount 

.25 274- .06 =4.2 1 1667 P. V. of ordinary annuity of I at 6% for 

5 periods 
4.211667X1,000=4,211.67 P. V. of ordinary annuity of 1000 

at 6% for 5 periods or amount to be paid as present 

value of first 5 rents. 

The sum to be paid for the last 5 rents of 
^1,200.00 each is the present value of an annuity 
of 5 rents at 6%, deferred 5 periods. An ordinary 
annuity of 5 rents of ^1,200.00 is computed thus: 

^ 4.211667X1200 = 5,054.00 P. V. of ordinary annuity. 
Then 5,054.00X7473 = 3,776.85 P.V. of annuity deferred 5 periods 

SUMMARY 

^4,211.67 present value of first 5 rents of ^1,000.00 each 
3,776.85 " " " last 5 rents of $1,200.00 " 

$7,988.52 total to be paid. 

Proof 

The ^7,988.52 would be charged to leasehold 
premium, and at the close of each year, journal 
entries would be made 



INTRODUCTION TO ACTUARIAL SCIENCE 87 

Debiting Rent (^1,000 first five years, ^1,200 

last five) 
Crediting Interest (6% of balance of lease- 
hold premium account) 
Crediting LeasehoM Premium (enough to 
balance the journal entry) 
If ^7,988.52 is the correct present value, the entry 
at the end of the fifth year should reduce the 
leasehold account to ^5,054.00, the present 
value as shown above of an ordinary annuity at 
6% of five rents of ^1,200 each, and the entry at 
the end of the tenth year should close the account. 

TABLE OF ENTRIES 



End of 


Debit 


Credit 


Credit 


Balance 


year 


rent 


interest 


leasehold 
premium 


leasehold 
premium 

7,988.52 


I 


1,000.00 


479.31 


520.69 


7,467.83 


2 


1,000.00 


448.07 


551.93 


6,915.90 


3 


1,000.00 


414-95 


585.05 


6,330.85 


•? 


1,000.00 


379.85 


620.15 


5,710.70 


5 


1,000.00 


342.64 


657.36 


5,053.34 



At this point a discrepancy appears. The 
balance should be ^5,054.00 and it is $5,053.34 
instead. The present value, .7453 is not suflS- 
ciently exact for the purposes of this problem; 
therefore the value of V^ @ 6% will be computed 
and the problem resolved: 



88 INTRODUCTION TO ACTUARIAL SCIENCE 



I -M .06 = .943 396226 
.943 3 96226 -M. 06 = .8 89996440 
.889996440-M.o6 = . 839619283 
.839619283 -r- 1.06 = 792093663 
.792093663 -M.o6=.747258i73 
periods hence 
1-.747258173 =.252741827 
.252741827-f- .06=4.2123637 
of I at 6% for 5 periods 
4.2123637X1000=4,212.3637 

of 1000 at 6% for 5 periods, or amount to be paid as 
present value of first 5 rents 
4.2123637X1200 = 5,054.83644 P.V. of ordinary annuity 
of 1200 at 6% for 5 periods 
5,054.83644X747258173 =3,777.2678 P. V. of deferred annuity 



P.V. of I at 6% due 5 

compound discount 
P.V. of ordinary annuity 

P.V. of ordinary annuity 





SUMMARY 






$4,212.3637 


P.V. of first 5 rents; $1,000 each 




3,777.2678 


" of last 5 rents; 1,200 " 




$7,989.6315 


total 

PROOF 






End of Debit 


Credit 


Credit 


Balance 


year rent 


interest 


leasehold 


leasehold 






premium 


premium 
7,989.63 


I 1,000.00 


479.38 


520.62 


7,469.01 


2 1,000.00 


448.14 


551.86 


6,917.15 


3 1,000.00 


415.03 


584.97 


6,332.18 


4 1,000.00 


379.93 


620.07 


5,712.11 


5 1,000.00 


342.73 


657.27 


5,054.84 


6 1,200.00 


303.29 


896.71 


4,158.13 


7 1,200.00 


249.49 


950.51 


3,207.62 


8 1,200.00 


192.46 


1,007.54 


2,200.08 


9 1,200.00 


132.00 


1,068.00 


1,132.08 


10 1,200.00 


67.92 


1,132.08 





11,000.00 


3,010.37 


7,989.6^ 





INTRODUCTION TO ACTUARIAL SCIENCE 89 

Problem lo 
(May 1920) 
The maximum sum insured under the sol- 
diers* insurance with the government is payable 
$57.50 per month for 20 years certain after death 
or disability. How can the equivalent sum 
payable in cash be found? Give the correct 
formula and explain why 12 x 57.50 x A^j differs 
from the equivalent cash sum. 
Given: A^^ at 3>^= 14.21 

Solution 10 

The problem is indefinite as to whether the 
first monthly payment is due immediately after 
death or disability, or one month thereafter. 
This is immaterial, however, if it is assumed 
that *^the equivalent sum payable in cash'' is 
payable on the same date that the first monthly 
payment would have been made. 

If the monthly payments were made they 
would constitute an annuity due of 240 rents of 
$57.50 each, payable monthly. The formula 
12 X 57.50 X a—^ is not correct for two reasons. 

In the first place, $14.21, the value of a— j at 
3/^%> is the present value of an ordinary 
annuity, while the annuity in question is an 
annuity due. That is, $14.21 is the present value 
of an annuity the rents of which are payable at 
the end of each period, while in this case the rents 



90 INTRODUCTION TO ACTUARIAL SCIENCE 

are payable at the beginning of each period, 
since it is assumed that the equivalent cash sum 
is payable on the date when the first monthly 
installment would have been due. 

In the second place, the formula 12 x 57.50 x 
^^ is based on the assumption that twenty 
annual payments are to be made, each of which 
is 12 X 57.50, or ^690. The facts are that 240 
monthly payments are to be made of ^57.50 
each. The annual rate is therefore only nominal; 
the effective rate is 3 J/2% divided by 12, or 7-24%. 
Therefore, instead of using <^— j at 3^2%^ ^^ 
14.21, it will be necessary to use the present value 
of an annuity due of 240 rents at 7-24%. The 
present value of an annuity due of 240 rents is 
computed by finding the present value of an 
ordinary annuity of 239 rents and adding i rent. 
This could be represented by a—^+i^the rate 
being 7-24%. The formula would be 57.50 x 

Problem 11 
(May 1920) 

The ^4% Victory notes mature at par on 

May 20, 1923. If a purchaser buys at ^96.20 

on May 20, 1920, calculate the approximate 

yield per cent. 

Given: 2^% 3% 

A6= 5-4624 5-4172 

V«= .8498 .837s 



INTRODUCTION TO ACTUARIAL SCIENCE 9I 

Solution II 

The approximate rate can be computed by 
determining what the price would have been if 
the bond had been purchased at an effective 
rate of 2^% per period of six months; also what 
the price would have been on a basis of 3% per 
period. The difference between these prices 
caused by an increase of 3^% in the rate will 
serve as an approximate measure of the excess 
of the effective rate over 2^% when the pur- 
chase was made at ^96.20. 



Price on a basis of 2^% per six months: 




Effective rate on par 


2.75 


Nominal " " " 


2.375 


Difference 


'2,7S 


Multiply by Pres. Val. of annuity at 2^% 


5.4624 


Discount 


2 . 0484 


Par 


100.00 


Discount 


2.05 


Price 


97.95 


Price on a basis of 3% per six months: 




Effective rate on par 


3.00 


Nominal " " " 


2.375 


Difference 


.625 


Multiply by Pres. Val. of annuity at 3% 


5.4172 


Discount 


3.38575 


Par 


100.00 


Discount 


3-39 


Price 


96.61 



92 INTRODUCTION TO ACTUARIAL SCIENCE 

At an effective rate of 2}i% the price would be 97-95 

" " " " "3 % " " " " 96.61 

An increase of X% reduces the price i . 34 

Price on basis of 2}i% per period 97-95 

unknown basis 96 . 20 



(( « 



Decrease in price i . 75 

Now if a decrease of ^1.34 in the price is caused 

by an increase of 3^% in the rate, the decrease 

of ^1.75 in the price is caused by an increase of 

I 7? 
approximately -^ of J^% or .3265%. 

1-34 

Then 2.75% plus .3265% = 3.0765%, the approx- 
imate rate per period, and 6.153% is the ap- 
proximate rate per annum. 

Schedule of Amortization 



Date 


Coupon 


Effective 
Income 


Discount 
Amortized 


Value 


May 20, 1920 








96.20 


Nov. 20, 1920 


2.38 


2.96 


.58 


96.78 


May 20, 1 92 1 


2.37 


2.98 


.61 


97-39 


Nov. 20, 1 92 1 


2.38 


3.00 


.62 


98.01 


May 20, 1922 


2.37 


3.02 


^^S 


98.66 


Nov. 20, 1922 


2.38 


3.04 


.66 


99.32 


May 20, 1923 


2.37 


3.06 


.69 


100.01 




14 25 


18.06 


3.81 





This schedule shows that the 3.0765% rate is 
an unusually close approximation. 



INTRODUCTION TO ACTUARIAL SCIENCE 93 

Problem 12 

(May 1920) 

A company is issuing ^100,000 of 4% 20-year 
bonds, which it wishes to pay at maturity by 
means of a sinking fund, in which equaFannual 
deposits are to be made. The board of directors 
wishes to assume that this fund will earn 53^% 
interest for the first five years, 5% for the next 
five years and 4% for the last ten years. What 
is the annual deposit required ? 



Given: 


S^% 


s% 


4% 


Sa 


S-S8i 


5.526 


S-4i6 


Sio 


12.87s 


12.578 


12.006 


(l+i)5 


1.307 


1.276 


1. 217 


(i+i)" 


1.708 


1.629 


1.480 




Solution 


12 





The problem states that the directors wish 
"'to assume that this fund will earn 5^2% 
interest for the first five years, 5% for the next 
five years and 4% for the last ten years." It is 
doubtful whether the examiners intended this 
statement to be interpreted literally. The fund 
will not earn sJ^% interest during the first five 
years unless the first despoit is made at the 
beginning of the first year. This is not cus- 
tomary and besides it would require a recom- 



94 INTRODUCTION TO ACTUARIAL SCIENCE 

putation of the various present values stated 
in the problem, converting them from present 
values of ordinary annuities to present values of 
annuities due. 

The problem will be solved first on the 
assumption that the deposits are made at the 
end of each year, in which case the fund will 
earn S/^% interest during the second, third, 
fourth and fifth years, 5% during the next five 
years, and 4% during the last ten years. It will 
then be solved on the basis of a literal inter- 
pretation of the statement of prospective interest 
earnings. 

It is regrettable that the values stated in the 
problem are not carried to six decimal places; 
the three place numbers are not exact enough to 
permit proving the correctness of the sinking 
fund contribution. In the solutions which fol- 
low, the first column shows the results obtained 
by using the three place values stated in the 
problem; the second column shows the results 
obtained by using six place values. 

Computation on assumption that contribu- 
tions to the fund are made at the end of each 
year: 



INTRODUCTION TO ACTUARIAL SCIENCE 95 

First five contributions: (3 place) (6 place) 



Amount at end of first five years 
Multiply by (1.05)^ 


5581 
1.276 


5.581091 
I. 27628 I 


Amt. of I St five contributions at 

end of loth year 
Multiply by (1.04)10 


7.121356 

1.480 


7.123040 
1.480244 


4.mount of 1st five contributions at 
end of 20th year 


10.539607 


10.543837 


Next five contributions: 






Amount at end of tenth year 
Multiply by (1.04)1° 


5.526 
1.480 


5.525631 

I . 480244 


Amount of 2nd five contributions at 
end 20 year 


8.17848 


8.179282 


Next ten contributions : 






Amount at end of 20th year 


12.006 


12.006107 



Summary: 

First five contributions amount to 10.539607 

Next five " " " 8.17848 

Next ten " " " 12.006 



10.543837 

8.179282 

12.006107 



Total 



30.724087 30.729226 



Sinking Fund Contribution: : 

100,000 divided by 30.724,087= 3,254.78 

100,000 " " 30.729,226= 3,254.23 

The following schedule of accumulation is 
prepared by way of proof, although the applicant 



96 INTRODUCTION TO ACTUARIAL SCIENCE 

is not required and would probably not have time 
to prepare it in the examination. 



Accumulation of Fund 



First 
Second 


Year Contribution 
" Interest at sH% 
Contribution 


3,254 78 

179.01 

3,254.78 


3,254.23 

178.98 

3,254.23 


Third 


Total 
" Interest at sH% 
Contribution 


6,688.57 

367.87 

3,254 78 


6,687 44 

367.81 
3,254.23 


Fourth 


Total 

" Interest at sH% 
Contribution 


10,311.22 

567.12 
3,254.78 


10,309.48 

567.02 

3,254.23 


Fifth 


Total 
" Interest at sH% 
Contribution 


14,133.12 

777.32 
3,254.78 


14,130.73 

777.19 

3,254.23 


Sixth 


Total 
" Interest at 5% 
Contribution 


18,165.22 

908.26 

3,254 78 


18,162.15 

908.11 

3,254.23 


Seventh 


Total 
" Interest at 5% 
Contribution 


22,328.26 
1,116.41 
3,254.78 


22,324.49 
1,116.22 

3,254.23 


Eighth 


Total 
" Interest at 5% 
Contribution 


26,699.45 

1,334.97 
3,254 78 


26,694.94 

1,334.75 
3,254.23 


Ninth 


Total 
" Interest at 5% 
Contribution 


31,289.20 
1,564.46 
3,254.78 


31,283.92 
1,564.20 
3,254.23 


Tenth 


Total 
" Interest at 5% 
Contribution 


36,108.44 
1,805.42 
3,254.78 


36,102.35 
1,805.12 
3,254.23 




Total 


41,168.64 


41,161.70 



INTRODUCTION TO ACTUARIAL SCIENCE 97 



Eleventh Year Interest at 4% 
Contribution 

Total 

Twelfth " Interest at 4% 
Contribution 

Total 
Thirteenth " Interest at 4% 
Contribution 

Total 
Fourteenth " Interest at 4% 
Contribution 

Total 
Fifteenth " Interest at 4% 
Contribution 

Total 
Sixteenth " Interest at 4% 
Contribution 

Total 

Seventeenth " Interest at 4% 
Contribution 

Total 
Eighteenth ** Interest at 4% 
Contribution 

Total 
Nineteenth " Interest at 4% 
Contribution 

Total 
Twentieth " Interest at 4% 
Contribution 

Total 



1,646.75 1,646.47 

3,254.78 3,254.23 



46,070.17 


46,062 . 40 


1,842.81 


1,842.50 


3,254.78 


3,254.23 


51,167.76 


51,159.13 


2,046.71 


2,046.37 


3,254-78 


3,254.23 



56,469.25 56,459.73 

2.258.77 2,258.39 

3.254.78 3,254.23 



61,982.80 61,972.35 

2,479.31 2,478.89 
3,254.78 3,254.23 



67,716.89 67,705.47 

2,708.68 2,708.22 

3,254.78 3,254.23 



73,680.35 73,667.92 
2,947.21 2,946.72 
3,254.78 3,254.23 



79,882.34 79,868.87 

3,195.29 3,194.75 
3,254.78 3,254.23 



86,332.41 86,317.85 

3,453.30 3,452.71 
3,254.78 3,254.23 



93,040.49 93,024.79 
3,721.62 3,720.99 
3,254.78 3,254.23 



100,016.89 100,000.01 



98 INTRODUCTION TO ACTUARIAL SCIENCE 

Computation on assumption that contribu- 
tions are made at the beginning of each year: 

First five contributions: (3 place) (6 place) 

Amount of ordinary annuity of i for 5 



periods at sH% 5 -581 
Multiply by 1055 


5.581091 
1.055 


Amount of annuity due of i for 5 periods 

at5>^% 5-887955 
Multiply by i .05^ i . 276 


5.888051 
I. 27628 I 


Amount of first 5 contributions at end of 

loth year 7513030 
Multiply by 1. 04IO 1.48 


7.514806 
1.480244 


Amount of first 5 contributions at end of 

20th year 11. i 19284 


II. 123750 


Next five contributions : 




Amount of ordinary annuity of i for 5 

periods at 5% 5-526 
Multiply by ^ 1.05 


5.525631 
1.05 


Amount of annuity due of i for 5 periods 

at 5% 5.8023 
Multiply by 1. 04I0 1.48 


5.801913 

I . 480244 


Amount of second 5 contributions at end 
of 20th year 8.587404 


8.588247 


Next ten contributions: (3 place) 


(6 place) 


Amount of ordinary annuity of i for 10 

periods at 4% 12.006 
Multiply by 1.04 


12.006107 
1.04 


Amount of annuity due of i for 10 periods 

at 4% 12.48624 


12.486351 



INTRODUCTION TO ACTUARIAL SCIENCE 99 



Summary: 



First five contributions amount to 
Next five " " " 

Next ten " " " 

Total 



I I. I 19284 II. 123750 

8.587404 8.588247 

12.48624 12.486351 



32.192928 32.198348 



Sinking Fund Contribution: 

100,000 divided by 32.192,928 = 
100,000 " " 32.198,348 = 

1st Year Contribution 

Interest at 5^% 
2nd " Contribution 



3rd 



4th 



5th 



6th 



7th 



3>io6.27 

3,106.27 

170.84 

3,106.27 



Total at interest 2nd year 


6,383.38 


Interest at sH% 


351.09 


Contribution 


3,106.27 


Total at interest 3rd year 


9,840.74 


Interest at sH% 


541.24 


Contribution 


3,106.27 


Total at interest 4th year 


13,488.25 


Interest at sH% 


741.85 


Contribution 


3,106.27 


Total at interest 5th year 


17,336.37 


Interest at 5>^% 


953.50 


Contribution 


3,106.27 


Total at interest 6th year 


21,396.14 


Interest at 5% 


1,069.81 


Contribution 


3,106.27 



Total at interest 7th year 25,572.22 
Interest at 5% 1,278.61 



3,105 


75 


3,105 
170 


75 

82 


3,105 


75 


6,382 
351 


32 
03 


3,105. 


75 


9,839 
541 


10 
15 


3,105 


75 


13,486 
741 


00 
73 


3,105 


75 


17,333 

953 


48 
34 


3,105 


7S 


21,392 
1,069 


57 
63 


3,105 


75 


25,567 
1,278 


95 
40 



lOO INTRODUCTION TO ACTUARIAL SCIENCE 

8th Year Contribution 3,106.27 3>io5.75 

Total at interest 8th year 29,957. 10 29,952. 10 

Interest at 5% 1,497.86 1,497.61 

9th " Contribution 3,106.27 3>io5.75 

Total at interest 9th year 34,561 . 23 34,555 .46 

Interest at 5% 1,728.06 1,727.77 

loth " Contribution 3,106.27 3,105.75 

Total at interest loth year 39,395.56 39,388.98 

Interest at 5 % i ,969 . 78 i ,969 . 45 

nth " Contribution 3,106.27 3,105.75 

" Total at interest nth year 44,471.61 44,464.18 

Interest at 4% 1,778 . 86 1,778 . 57 

I2th " Contribution 3,106.27 3,105.75 

Total at interest 12th year 49,356.74 49,348. 50 

Interest at 4% 1,974 . 27 1,973 . 94 

13 th " Contribution 3,106.27 3,105.75 

Total at interest 13th year 54,437.28 54,428. 19 

Interest at 4% 2, 1 77 . 49 2, 1 77 . 1 3 

14th *' Contribution 3,106.27 3,105.75 

Total at interest 14th year 59,721.04 59,711.07 

Interest at 4% 2,388.84 2,388.44 

15th *' Contribution 3,106.27 3,105.75 

Total at interest 15th year 65,216. 15 65,205 .26 

Interest at 4% 2,608 . 65 2,608 . 21 

i6th " Contribution 3,106.27 3,105.75 

Total at interest i6th year 70,931.07 70,919.22 

Interest at 4% 2,837.24 2,836.77 

17th " Contribution 3,106.27 3,105.75 

Total at interest 17th year 76,874.58 76,861.74 

Interest at 4% 3,074 . 98 3,074 . 47 



INTRODUCTION TO ACTUARIAL SCIENCE lOI 

i8th Year Contribution 3,106.27 3>I05.7S 

Total at interest 1 8th year 83,055.83 83,041.96 

Interest at 4% 3,322.23 3,321.68 

19th " Contribution 3,106.27 3>io5.75 

Total at interest 19th year 89,484.33 89,469.39 

Interest at 4% 3,579 . 37 3,578 . 78 

20th ** Contribution 3,106.27 3*105.75 

Total at interest 20th year 96,169.97 96,153.92 

Interest at 4% 3,846.80 3,846.16 

Total 100,016.77 100,000.08 



9 



